UC-NRLF 


I  Hill  1 1  III 


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08. 

LI  BR  ARY 

OF   THK 

UNIVERSITY  OF  CALIFORNIA 

Received __._^_____^/C-/:^.....,  i88p^ 
Accessions  No'^/y^/S        Shelf  No. 


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httD://www.archive.ora/details/culminationofsciOOsmitrich 


THE   CULMINATIOK 


OF    THE 


SCIENCE  OF  LOGIC 


WITH   SYNOPSES   OF 


A.LL  POSSIBLE  VALID  FORMS  OF  CATEGORICAL  REASONING  IN 
SYLLOGISMS  OF  BOTH  THR6e  AND  FOUR  TERMS. 


BY 


JOHN    C.    SMITH, 

A  MEMBER  OP  THE  BROOKLYN  BAR. 


-    OF  THE 

PUBLISHED  BY 

HERBERT   C.  SMITH, 
16  COURT  STREET,   BROOKLYN.   N.  Y. 


Copyright,  1888,  by  Herbert  C.  Smith. 


>*7//j 


Electrotyped  by  R.  Harmer  Smith  &  Sons.  82  Beelsiiian  St..  New  YorK. 


PREFACE. 


The  following  are  two  chapters  of  a  treatise  now  in 
course  of  preparation,  and  to  be  entitled  ''Logic  as  a 
Pure  Science,  illustrated  only  by  means  of  symbols  indefi- 
nite in  material,  but  definite  in  logical  signification,  with, 
synopses  of  all  possible  valid  forms  of  categorical  reason- 
ing in  syllogisms  of  both  three  and  four  terms." 

The  preparation  of  the  treatise  was  undertaken  with 
but  little  expectation  that  it,  or  any  part  of  it,  would 
ever  be  published  ;  and  certainly,  with  no  thought  of  its 
resulting  in  any  new  contribution  to  the  science. 

The  author  had  long  thought  an  elementary  treatise 
on  Logic  as  a  pure  science,  with  matter  wholly  elimi- 
nated, a  desideratum  ;  and  if  any  such  has  ever  appeared, 
he  is  not  aware  of  it.  He  acknowledges,  however,  that 
his  acquaintance  with  the  literature  of  the  science  is 
very  limited.  In  writing  the  treatise,  therefore,  no  con- 
crete examples  were  employed,  but  only  those  with  sym- 
bols indefinite  as  to  matter,  but  made  definite  as  to 
their  logical  signification. 

The  symbols  adopted  w^ere  the  letters  N,  D,  and  J,  to 


IV  PREFACE. 

represent  the  Minor,  Middle  and  Major  terms  of  the  Syl- 
logism ;  they  being  the  middle  letters  of  these  words 
respectively.  S,  M  and  P  are  usually  employed,  as  the 
initials  of  Subject,  Middle  and  Predicate,  but  S  and  P 
are  objectionable,  being  equally  applicable  to  the  sub- 
ject and  predicate  of  the  premises  (as  propositions),  in 
each  of  which  but  one  occurs  in  the  statement  of  Syllo- 
gisms, and  that  one  in  its  appropriate  place  in  such 
representation  in  both  premises,  only  in  Syllogisms  in 
the  first  figure ;  in  one  premise  only,  in  the  second  and 
third  figures ;  and  in  neither,  in  the  fourth ;  and  their 
dual  possible  representations  tend  to  confusion.  Dis- 
tribution and  non-distribution  are  signified  by  the  use 
of  capitals  to  represent  terms  distributed,  and  small 
letters,  terms  not  distributed.  JSTegation,  in  universal 
propositions,  is  indicated  by  crossing  the  capital  let- 
ter representing  the  subject.  The  copula  is  expressed 
by  the  characters,  ^'  — "  for  "is,"  and  "--"  for 
''is  not." 

In  translating  the  symbols  and  characters  as  em- 
ployed in  propositions  into  spoken  language,  the  sig- 
nification of  the  symbols  should  of  course  be  expressed 
in  respect  to  the  subject,  but  implied  in  respect  to  the 
predicate,  according  to  common  usage  and  the  well- 
known  rules  that  all  universal  propositions  (and  no  par- 
ticular) distribute  the  subject,  and  all  negative  (and  no 
afiirmative)  the  predicate. 


PREFACE.  V 

Thus  the  four  propositions,  A,  E,  I,  O,  when  written 
in  symbols  and  characters  as  above,  should  be  read  and 
understood  as  follows : 

(A)  D  —  j  All  D  is  j  {meaning  All  D  is  some  J) 

(E)  &— J  No  D  is  J  (       ''       No  D  is  «W2/  J) 

(I)  d  —  j  Some  D  is  j  (        ''       Some  D  is  some  J) 

(0)  d — ^J  Some  D  is  not  J  (       ^'       Some  D  is  not  awy  J) 

The  consideration  of  Hypotheticals  was  reached  in 
the  preparation  of  the  treatise,  and  in  the  course  thereof, 
analyses  of  conditional  propositions  of  both  three  and 
four  terms,  in  all  forms  in  which  they  can  be  expressed, 
were  made ;  and  the  study  of  their  results  led  to 
the  gradual  unfolding  of  the  doctrine  of  Sorites  con- 
tained in  the  second  of  the  following  chapters. 

That  doctrine  is  the  culmination  of  the  Science  of 
Logic,  which  without  it  has  hitherto  been  incomplete. 

The  treatise,  up  to  this  point,  had  been  written 
wholly  in  short-hand,  and  to  guard  against  the  possi- 
bility that  the  discovery  might  be  lost  if  the  author 
should  not  live  to  finish  it,  and  the  notes  should  not  be 
deciphered,  these  chapters  were  written  out  in  full,  and 
put  in  position  where  they  would  be  found  and  pub- 
lished, in  such  contingency. 

But,  inasmuch  as  the  work  yet  remains  to  be  com- 
pleted, and  the  notes  to  be  written  out  (which  can  only 
be  done  by  the  author,  his  system  of  short-hand  being  in 
many  respects  peculiar),  its  appearance  will  be  consider- 


Vi  PREFACE. 

ably  delayed  ;  and  as  the  discovery,  when  made  known, 
will,  it  is  believed,  not  only  be  an  occasion  of  interest 
from  a  scientific  point  of  view,  but  will  prove  also  to 
be  of  practical  utility,  the  author  has  determined  to 
publish  these  two  chapters  in  advance.  The  chapter 
on  Enthymemes  is  published  as  preliminary,  and  to  ex- 
hibit the  synopses  therein  contained  (of  which  the  last 
shows  all  valid  simple  Syllogisms  [of  three  terms]  at  full 
length  and  in  regular  form),  in  connection  with  those 
contained  in  the  chapter  on  Sorites  (Syllogisms  of  four 
terms),  thus  bringing  together,  as  it  were  in  one  view, 
all  possible  valid  forms  of  categorical  reasoning.  To 
those  for  whose  benefit  they  are  thus  published  the 
chapters  may  seem  to  be  unnecessarily  diffuse  and 
minute,  but  to  condense  them  would  involve  very  con- 
siderable labor,  and  they  are  therefore  put  forth  in  the 
form  in  which  they  were  written  to  take  their  appro- 
priate places  in  the  full  treatise,  trusting  that  their 
minor  defects  and  redundancies  may  be  overlooked. 

If  the  remainder  of  the  treatise  shall  never  appear 
from  the  author's  pen,  there  will  be  little  or  nothing 
lost.  The  suggestion  herein  made,  if  it  have  any  merit, 
will  lead  other  and  abler  pens  to  supply  the  desideratum. 

Brooklyn,  January  14,  18S8. 


UITIVBIISITY] 

OF    ENTHYMEMES. 

§  1.  We  have  hitherto  considered  the  process  of 
reasoning  with  three  terms,  categorically,  in  its  full 
expression,  and  have  examined  all  the  possible  forms 
of  such  expression.  Such  forms  are  seldom  resorted 
to,  either  in  common  conversation  or  formal  discourse, 
whether  spoken  or  written,  but  abridged  forms  of  argu- 
ment are  employed  in  which  only  part  of  the  process 
is  expressed,  the  remainder  being  implied,  and  being 
usually  so  obvious  as  not  to  require  expression.  AVe 
come  now  to  consider  sucli  abridged  forms. 

They  are  called  Enthymemes. 

§  2.  An  Enthymeme  is  a  Syllogism  of  which  but  two 
propositions  are  expressed,  the  third  being  implied. 

Enthymemes  are  of  three  orders  ; 

1st.  That  in  which  the  major  premise  is  implied. 
2d.  That  in  which  the  minor  premise  is  implied. 
3d.   That  in  which  the  conclusion  is  implied. 

The  following  are  examples. 

Of  the  first : 

N  -  d; 
.-.  X  -  j. 


Of  the  second : 


D-j; 

••  N  -  j. 


8  LOGIC  AS  A  PURE  SCIENCE. 

Of  the  third : 

D  -  ], 
and  N  —  d. 

In  each  case  the  three  terms  requisite  to  make  up 
a  full  Syllogism  appear,  and  the  implied  premise  or  con- 
clusion can  be  readily  supplied. 

Enthymemes  of  the  first  order  are  herein  called 
Minor,  and  those  of  the  second  order  Major  Enthy- 
memes, from  the  names  of  their  expressed  premises, 
respectively. 

§  3.  As  every  Enthymeme,  together  with  its  implied 
premise  or  conclusion,  is  a  Syllogism,  it  is  evident  that 
only  such  can  be  valid  as  are  symbolized  by  the  letters 
by  which  the  expressed  propositions  are  symbolized,  in 
the  combinations  of  vowels  symbolizing  the  propositions 
of  all  allowable  moods  of  categorical  syllogisms,  as 
hereinbefore  shown. 

By  reference  thereto,  it  will  be  found  that  all  valid 
Enthymemes  must  consist  of  propositions  of  which  the 
following  are  the  symbols  ;  namely. 


Of  the  first  order. 
{Minor  Enthymemes.) 

— ,  A,  A ; 

—  A,  B; 
-A,  I; 

—  A,  O; 
— ,  E,  E; 

—  E,  0; 
-,  I,  I; 
-,  I,  0 ; 
-,  0,  0. 


Of  the  second  order. 
{Major  Erithymemes.) 

A, -A 
A,—,  B 
A,-,  I 
A,  -  0 
E,  -  E 
E,  -  0 
I,  -,  I 

0,  -,  0. 


Of  the  third  order. 


A,  A,- 
A,  B,  — 
A,  I,  - 

A,  0,  - 

B,  A,  - 
E.  I,  — 
I,  A,- 

0,  A,  -. 


ENTHYMEMES.  9 

The  symbols  of  minor  and  major  Enthymemes  are  the 
same,  except  that  there  is  no  valid  major  Enthymeme  in 
I,  O.  There  are  no  valid  minors  in  E,  O,  except  useless 
ones.  Leaving  the  latter  out  of  consideration,  it  will  be 
found  that  A  occurs  four  times  as  the  symbol  of  the  pre- 
mise, and  but  once  as  the  symbol  of  the  conclusion  in 
both  minor  and  major  Enthymemes ;  E  once  in  minors 
and  twice  in  majors  as  the  symbol  of  the  premise,  and 
twice  in  each  as  the  symbol  of  the  conclusion  ;  I  twice  in 
minors  and  once  in  majors  as  the  symbol  of  the  premise 
and  twice  in  each  as  the  symbol  of  the  conclusion  ;  and 
O  once  as  the  symbol  of  the  premise  and  three  times  as 
the  symbol  of  the  conclusion  in  both  minors  and  majors. 

Minor  Enthymemes  are  the  most  common,  the  sup- 
pressed major  premise  being  usually  a  geneml  rule, 
readily  recognized  and  acquiesced  in  without  being 
expressed. 

Enthymemes  of  the  third  order  are  seldom  employed,, 
except  in  combination  with  one  of  the  first  or  second 
order.  They  will  be  referred  to  when  we  come  to  the  con- 
sideration of  Sorites,  and  it  will  be  found  that  they  occur 
sometimes  in  the  order  of  the  symbols  above  shown, 
namely,  major  premise  first,  and  minor  second  ;  and  some- 
times in  the  reverse  order,  minor  first,  and  major  second. 

§  4.  To  the  three  orders  may  be  added  a  fourth  ;  viz., 
an  Enthymeme  with  but  one  expressed  and  two  implied 
propositions.  Every  demonstrable  categorical  proposi- 
tion, put  forth  independently  as  the  expression  of  a 
judgment,  is  such  an  Enthymeme,  being  the  conclusion 
of  two  implied  premises.  If  the  question  is  asked, 
''What  is  N?"  the  answer  must  be  either  a  random 


10  LOGIC   AS  A   PURE   SCIET^CE. 

expression  in  the  form  of  a  proposition,  but  meaningless, 
or  the  result  of  thought  more  or  less  deliberate,  and 
therefore  based  upon  some  reason,  which,  as  we  have 
before  seen,  is  a  just  (or  assumed  as  just)  ground  of  con- 
clusion. This  ground  must  be  a  mental  comparison  of 
the  subject,  IN",  with  some  other  term,  and  of  that  again 
with  the  term  predicated  of  the  subject  in  the  answer. 
The  premises  thus  formed,  but  not  expressed,  must  be 
obvious  to  the  questioner,  when  the  answer  is  given,  and 
therefore  admitted ;  or  otherwise  explanation  would  be 
demanded.  Were  this  not  so,  there  could  be  no  reason- 
ing without  going  back  in  every  process  to  some  inde- 
monstrable proposition  (axiom  or  postulate),  or  to  the 
Great  First  Cause;  mth  which  or  with  Whom,  when 
reached  in  the  process  of  investigation,  we  must  necessa- 
rily set  out  in  retracing  our  steps  in  the  deductive  pro- 
cess of  reasoning. 

Such  an  Enthymeme  may  also  consist,  in  so  far  as  it  is 
expressed,  of  a  single  proposition  put  forth  as  a  premise 
(usually  the  major),  the  unexpressed  premise  and  conclu- 
sion being  left  to  be  gathered  from  the  attending  circum- 
stances or  from  the  subject-matter  under  consideration. 

§  5.  The  middle  term  will  of  course  be  that  term  of 
the  expressed  premise,  in  minor  and  major  Enthy- 
memes,  which  is-  not  common  to  both  propositions,  and 
in  Enthymemes  of  the  third  order,  that  which  is  common 
to  both ;  and  will  vary  in  position  according  to  the  fig- 
ure, and  the  character  of  the  premise,  whether  minor  or 
major.  In  minor  and  major  Enthymemes  it  may  or  may 
not  be  distributed,  according  to  the  mood,  and  character  of 
the  premise,  whether  minor  or  major  ;  but  in  Enthymemes 
of  the  third  order  must  be  at  least  once  distributed. 


ENTHYMEMES.  11 

§  6.  It  is  manifest,  that  there  are  three,  and  can  be  but 
three,  Enthymemes  ha\ing  two  expressed  propositions, 
viz.,  one  minor,  one  major,  and  one  of  the  third  order, 
in  each  allowable  mood  of  the  syllogism ;  and  as  the 
number  of  such  moods  is  twenty-four,  including  the  use- 
less ones,  so  the  number  of  Enthymemes  of  each  kind  is 
limited  to  twenty-four. 

The  follovv'ing  are  synopses  of  all  possible  valid  forms 
of  categorical  Enthymemes  of  two  expressed  propositions, 
together  with  the  implied  premise  or  conclusion  of  each, 
as  the  case  may  be.  On  the  first  page  of  each  of  the  two 
synoiDses  of  minor  and  major  Enthymemes  the  forms  of 
the  expressed  propositions  are  printed  in  full,  each  but 
once,  in  the  order  A,  I,  E,  O,  of  the  symbols  of  the  con- 
clusion, but  on  the  second  page  they  are  printed  in  full 
throughout.  Where  they  are  repeated,  they  will  be 
found  to  have  in  each  case  a  different  implied  proposi- 
tion. By  counting,  it  will  be  found  that  there  are  fifteen 
forms  of  the  expressed  propositions  of  minor  Enthy- 
memes (of  which  four  are  useless)  and  twelve  of  majors. 
The  capital  letters  in  the  names  of  the  moods  on  each 
page  of  the  synopses  are  the  symbols  of  the  proposition 
or  propositions  in  the  column  next  adjoining. 

The  synopsis  of  Enthymemes  of  the  third  order,  will 
serve  also  as  a  synopsis  of  those  of  the  fourth  order,  as 
first  described,  by  considering  the  words  *' expressed" 
and  ''implied"  as  transposed  in  the  headings  over  the 
columns  of  the  propositions. 

As  arranged  on  page  17,  and  read  across  the  page,  it 
exhibits  all  possible  valid  forms  of  categorical  reasoning 
with  three  terms,  at  full  length  and  in  regular  form,  in 
the  order  of  the  Moods  of  the  Syllogism. 


12 


LOGIC    AS   A   PUEE   SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms  of  Categorical  Enthymemes 
of  the  First  Order.     (Minors.) 

IN  THE  ORDER  A.  I,  E,  0,  OF  THE  SYMBOLS  OF  THE  CONCLUSIONS. 


Expressed  Propositions. 

Implied  Proposition. 

Moods  op 

III 

-Minor  Premise.          Conclusion. 

Major  Premise. 

Stllogisms. 

A,  A. 

N-  d; 

.-.     N-j. 

V     D-j. 

bArbara. 

A,  I. 

N-  d; 

•••      n  -  j. 

V     D-j. 

A,  a.  i.    Ist  fig. 

D-  n; 

•••      n  -  j. 

V     D-j. 

dArapti. 

tt 

// 

V      d-j. 

disamis. 

>; 

n 

•••     J-d. 

V         j-d. 

brAmantip. 
dimaris. 

I,    I. 

n-d; 

.-.      n  -  j. 

V     D-j. 

dAni. 

d-n; 

•••      n  -  j- 

V     D-j. 

dAtisi. 

A,E. 

N-  d; 

•.•     D-J. 
•.•     J-D. 

cElarent. 
cEsare. 

E,  E. 

?^-D 

;     .-.     N-J. 

•.•     J-d. 

cAmsstres. 

&-N 

;     .-.     N-J. 

•.•     J  -  d. 

cAmenes. 

A,  0. 

X-  d 

;     .-.      n  -^  J. 

V     D-J. 
•.•      J-D. 

E,  a,  0.    1st  fig. 
E,  a,  0.    2d  fig. 

D-n; 

II 

.-.      n  -^  J. 

V     D-J. 

•.•      d  -^  J. 

fElapton. 
bOkardo. 

II 

// 

•.•     J-D. 

fEsapo. 

E,  0. 

N-D 

;     .-.     n  -^  J. 

•.•     J-d. 

A,  6.  0.     2d  fig. 

D-N 

;     .-.      n-^J. 

•.•     J-d. 

A,€,o.   4th  fig. 

I,  0. 

n  -  d 

;     .-.      n  -w  J. 

V     D-J. 

fEHo. 

II 

/; 

V     J-D. 

fEstino. 

d-n 

;      .-.      n  -w  J. 

V     D-J. 

fEriso. 

// 

// 

•.•     J-D. 

frEsimn. 

0,  0. 

n-^D 

;     .-.      n  — -  J. 

•.•     J-d. 

bAroho. 

ENTHYMEMES. 


13 


Synopsis  of  all  Possible  Valid  Forms  of  Categorical  Enthymemes 
of  the  First  Order.    (Minors.) 

IN  THE  ORDER  OF  THE  MOODS  OF  CATEGORICAL  SYLLOGISMS. 


Moods  op 

KXPBESSBB  PBOFOSITIOHS. 

iMPLrCD  PBOPOSmON. 

Syixogisms. 

Minor  Premise. 

Conclusion, 

Major  Premise. 

harbAr-A. 

X-  d; 

.., 

^^  -  J. 

V       D  -  j. 

cdArEnt. 

X  -  d  ; 

.*. 

X-J. 

V       B-J. 

a,  A.  I. 
darll. 

X-  d  ; 
n  -  d  ; 

•'• 

n-j.) 
n  -  J.  i 

••        D  -  j. 

e.  A,  0. 
ferlO. 

X-  d; 
n  -  d  ; 

.*. 

•.•       ^  -  J. 

ce^ArE. 

X-  d; 

.-. 

X-J. 

•.•       J  -  D. 

camEstrEs. 

5?-D; 

.-, 

X-J. 

•••       J  -  d. 

e,  A.  0. 
festin  0. 

X  -  d  : 
n  -  d  ; 

,*, 

n-^J.  1 

n-^J.  (" 

•.•       J  -  D. 

a,  E,  0. 
barOkO 

X-D; 

n-^D; 

,*, 

n-^J. 
n-^J.  ■ 

•.•       J  -  d. 

darAptl. 

D  -  n  ; 

.-. 

n  -  J- 

•.•       D  -  j. 

disAmls. 

D-  n  ; 

/. 

n  -  j- 

•.•       d  -  j. 

datlsl. 

d  -  n  : 

.-. 

^  -  j- 

V       D  -  j. 

feiAptOn. 

D-  n; 

.-. 

n-^J. 

•.•       B-J. 

hokArdO. 

D-  n  ; 

.'. 

n-^J. 

V        d-^J. 

ferlsO. 

d  -  n  ; 

.*. 

n-^J. 

•.•       ^  -  J. 

bramAntlp. 

D-  n; 

.*. 

^  -  j- 

•.•        J  -  d. 

camEnEs. 
a,  E.  0. 

B-X; 
&-X; 

' , 

X-J.) 
n-^J.  S 

•.•        J  -  d. 

dimArls. 

D  -  n  ; 

.', 

^  -  j- 

•••        .]  -  d. 

fesApO. 

D-  n; 

.'. 

n-^J. 

V       J  -  D. 

f  regis  On. 

d  -  n  ; 

•*• 

n^J. 

•.•       J  -  D. 

14 


LOGIC   AS   A  PURE   SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms  of  Categorical  Enthymemes 
of  the  Second  Order.    (Majors.) 

IN  THE  ORDER  A,  I,  E,  0,  OF  THE  SYMBOLS  OF  THE  CONCLUSIONS. 


'*ii 

IH 

EXPBESSED  Propositions. 

Impt.tkt)  Proposition. 

Moods  of 

m 

Major  Premise.          Conclusion. 

Minor  Premise. 

Syllogisms. 

A,  A. 

D-j; 

.-.     N-j. 

V    K-d. 

barbAra. 

A,  I. 

D-i; 

.-.      n  -  j. 

V    K-d. 

a,  A,  L  1st  fig. 

// 

II 

•.•     11  -  d. 

darli. 

/' 

II 

V    D-ii. 

darAptl. 

It 

n 

V     d-n. 

datlsi. 

J-d; 

.-.      n  -  j. 

V    D-n. 

bramAntip. 

I,      1. 

d-j; 

.-.      n  -  j. 

•.•    D-n. 

disAmis. 

J-d; 

.-.      n  -  j. 

•••    D-n. 

ditnAris. 

A,  E. 

J-d; 

.-.     N  -  J. 

V    5^-D. 

camEstres. 

// 

'/ 

•.•    D-N. 

camEnes. 

E,  E. 

^-   J 

.-.     ?^  -  J. 

•.•  ]sr  -  d. 

cdArent. 

J-D 

,     .-.     J^-J. 

•.•    N  -  d. 

cesAre. 

A,  0. 

J-d, 

tt 

.-.      n-^J. 

II 

V     n-wD. 

a,  E,  0.  2d  fig. 
barOko. 

ti 

II 

V    D-N. 

a,  E.  0.  4th  fig.. 

E,  0. 

&-J; 

,'.      n  -^  J. 

•.•    N  -  d. 

e.  A,  0.  1st  fig. 

II 

ri 

•.•     n  -  d. 

ferlo. 

II 

It 

•.•    D  -  n. 

felApton. 

II 

tt 

•.•     d-n. 

ferlso. 

J-D; 

.-.      n->^J. 

V    N-d. 

e,  A,  0.  2d  fig. 

// 

II 

V     n-d. 

festlno. 

It 

11 

•.•    D-n. 

fesApo. 

n 

It 

•.•     d  -  n. 

freslson. 

0,0. 

d-wJ; 

.-.      n-^J. 

V    D-n. 

boTcArdo. 

ENTHYMEMES. 


15 


Synopsis  of  all  Possible  Valid  Forms  of  Categorical  Enthymemes 
of  the  Second  Order.     (Majors.) 

IN  THE  ORDER  OF  THE  MOODS  OF  CATEGORICAL  SYLLOGISMS. 


Moods  of 

Expressed  Propositions. 

Implied  Proposition. 

Syllogisms. 

Major  Premise. 

Conclusion. 

Minor  Premise. 

bArbarA. 

i     D-J; 

... 

^^  -  j. 

... 

X-d. 

cElarEnt. 

fi-- J; 

.*. 

?^-  J. 

... 

N-d. 

A,  a,  I. 

D-  j; 

.-. 

11  -  j- 

... 

]^-d. 

dAHZ. 

D-J; 

.*. 

11  -  j- 

•/ 

n- d. 

E,  a,  0. 

]^- J; 

.-. 

11-^  J. 

•.. 

N-d. 

fEriO. 

B-  J; 

.-. 

11-^  J. 

..• 

n  -  d. 

cEsarE. 

,f-D; 

.-. 

5^- J. 

... 

X-d. 

cAmestrEs. 

J-d; 

.-. 

2^- J. 

... 

?^-D. 

E,  a,  0. 

a=  -D; 

.*. 

n^J. 

•.• 

N- d. 

fEstinO. 

J-D; 

.-. 

n-^J. 

•/ 

n-d. 

A,  €,  0. 

J-d; 

.'. 

n^J. 

".' 

??-D. 

bArokO. 

J-d; 

.*. 

n  -^  J. 

*.* 

n-^D.. 

dAraptl. 

D  -  j  ; 

/. 

n- j. 

... 

D-n. 

disamls. 

d  -  j  : 

.-. 

n- j. 

•/ 

D-n. 

dAtisI. 

D  -  j  ; 

.-. 

n-J. 

•.' 

d-n. 

fElaptOn. 

&- J; 

/. 

n  -^  J. 

'.* 

D-n. 

bOkardO. 

d-^J; 

.*. 

n-^J. 

*,* 

D-n. 

fErisO. 

&- J; 

.-. 

n-^J. 

'.' 

d-n. 

brAmantlp. 

J-d; 

.*. 

n- j. 

•.• 

D-n. 

cAmenEs. 

J-d; 

.-. 

^-J. 

'.• 

D-I^. 

A,  €,  0. 

J-d; 

.*. 

n-wJ. 

".' 

B--]^. 

dimarls. 

j-d; 

.*. 

n  -  .]'• 

•.* 

D-n. 

fEsapO. 

J-D; 

.-. 

n-^J. 

•.• 

D-n. 

frEsisOn. 

J-D; 

••• 

n-^J. 

••• 

d-n. 

16 


LOGIC   AS   A   PUEE  SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms  of  Categorical  Enthymemes 
of  the  Third  Order. 

IN  THE  ORDER  A,  I,  E,  0,  OF  THE  SYMBOLS  OF  THE  CONCLUSIONS. 


)LS  OF 
E88ED 
IITION8. 

E2CPRESSED  Propositions. 

Implied  Proposition. 

Moods  of 

^«l 

Major  Premise.    Minor  Premise. 

Conclusion. 

Syllogisms. 

A,  A. 

D  —  j,  and  A'  —  d. 

.-.     N-j. 

harbarA. 

A,  A. 

I)  _  j,     .     X  -  d. 

.-.     n  -  j. 

a,  a,  /.    1st  fig. 

A,  I. 

D  -  j,     "      n  -  d. 

.-. 

daril. 

A,  A. 

D  -  j,     "     D  -  n. 

" 

daraptl. 

A,  I. 

D  -  j,     "      d  -  n. 

.*.           II 

datul. 

I,  A. 

d  _  j,     n     D  -  n. 

.'.          f 

disamls. 

A,  A. 

J  _  d,     "     D  —  n. 

.*.          " 

bramantlp. 

I,  A. 

j  _  d,     "     D  -  n. 

.*.          " 

dimarls. 

E,  A. 

©  _  J,      n      N  -  d. 

.-.     ^-J. 

celarEnt. 

B,  A. 

J  _  D,    "     N  -  d. 

" 

cesarE. 

A,  E. 

J  _  d,     "     5^-1). 

.'.          II 

camestrEs. 

A,  E. 

J  _  d,     "     ©•  -  N. 

II 

camenEs. 

E,  A. 

©  _  J,     ,/     N  -  d. 

.-.     n-^J. 

e,  a,  0.    1st  fig. 

E,  I. 

B-  —  J,     "      n  —  d. 

" 

fenO. 

E,  A. 

©  _  J,     n     D  -  n. 

ti 

felapiOn. 

E,  I. 

B-_J,     n     d  -  n. 

" 

ferisO. 

O,  A. 

d-^J,     "     D-n. 

,           II 

bokardO. 

A,  E. 

J  _  d,     "     N  -  D. 

II 

a,  e,  0.    2d  fig. 

A,  0. 

J  -  d,     "     n-_D. 

.           /' 

barokO. 

A,  E. 

J  _  d,     "     B  -  N. 

// 

a,  €,  0.   4th  fig. 

E,  A. 

J  _  D,    "     N  -  d. 

ti 

e,  a,  0.     Sd  fig. 

E,  I. 

J  _  D,    "      n  —  d. 

.           It 

festinO. 

E,  A. 

dF  —  D,    "     D  —  n. 

II 

fesapO. 

E,  I. 

a=  _  B,    "      d  —  n. 

" 

f  rests  On. 

ENTHY3IEMES. 


17 


Synopsis  of  all  Possible  Valid  Forms  of  Categorical  Enthymemes 
of  the  Third  Order. 

IN  THE  ORDER  OF  THE  MOODS  OF  CATEGORICAL  SYLLOGISMS. 


Moods  of 

Expressed  Propositioks.                      bcpi 

UED  Proposition, 

Syllogisms. 

Major  Premise.                 Minor  Premise. 

CoDclusiou, 

hArbAra. 

D  -  j,       and      X  -  d. 

•.     X-j. 

cElArent. 

3&  -  J,        "        X  -  d. 

•.     X-J. 

A.  A.  i. 

D-j,         "        X-d. 

••      n  -  j. 

dArli. 

D  -  j.         "         n  -  d. 

••      n  -  j. 

E,  A.  0. 

&  -  J.        "        X-d. 

•.     n-wj. 

jErlo. 

B-  -  J.        "        n  -  d. 

•.     n-«^J. 

cEsAre. 

^  _  D,       "        X-d. 

•.     X-J. 

cAmEgtres. 

J  _  d,       "        X-D. 

•.     X-J. 

E,  A,  0. 

^  _  I).       r        x-d. 

•.      n-^J. 

fE-<fI,-n. 

a:  _  D.       '         n  -  d. 

•.      n  -^  J. 

A.  E.  c . 

J  _  d,       "        X-D. 

•.      n-^J, 

hArOko. 

J-d.       "         n-^D. 

•,      n-^J. 

lArApti. 

D-j,        '.        D-n. 

•.     u  -  j. 

aisAyni?. 

d-j,        ■'        D  -  n. 

••      n  -  j. 

dAtlsi. 

D  -  j,,        n        d  -  n. 

••      n  -  .j- 

fElApton. 

D-J,        ''        D  -  n. 

•.      n-^J. 

bOkArdo. 

d  -^J.        "        D  —  n. 

•.      n-^J. 

rErlso. 

D  -  J,        "        d  -  n. 

•.      n-^J. 

brAniAntip. 

J  _  d,,       r.        D  -  n. 

••      1^  -  j- 

cA?nE>tes. 

J  _  d,        n        D  -  X. 

•.     5f-J. 

A.  E.  0. 

J  _  d,        "        D  -  X. 

•.      n-^J. 

dImAni. 

j  _  d,,       "        D  —  n. 

••      n  -  j. 

TEsApo. 

J  _  D,       '         D  -  n. 

•.     n  -^  J. 

rEfilson. 

J  _  D,       "         d  -  n. 

•.      n-^J. 

18  LOGIC  AS  A   PURE   SCIENCE. 

§  7.  The  following  will  serve  as  rules  by  which  the 
implied  proposition  of  every  Enthymeme  having  two 
expressed  propositions  may  be  supplied,  the  first  being 
applicable  to  those  of  either  the  first  or  second  order, 
and  the  second  to  those  of  the  third. 

1st.  The  term  of  the  conclusion  of  an  Enthymeme 
of  either  the  first  or  second  order  which  is 
common  to  both  expressed  propositions  deter- 
mines the  character  of  the  expressed  premise, 
whether  minor  or  major,  according  as  the 
same  is  either  the  subject  {minor  term)  or 
predicate  (major  term)  of  the  conclusion,  and 
the  implied  premise  may  be  found  by  com- 
paring the  other  two  terms. 
2d.  The  term  of  the  expressed  minor  premise  of 
an  Enthymeme  of  the  third  order  not  common 
to  both  premises  is  the  subject,  and  that  of 
the  expressed  major  premise  not  common  to 
both  is  the  predicate,  of  the  implied  conclu- 
sion, which  is  universal  or  particular,  and 
affirmative  or  negative,  as  called  for  by  the 
premises. 


OF    SORITES. 

§  1.  We  come  now  to  the  consideration  of  reasoning 
with  four  terms,  categorically;  and  we  shall  herein- 
after find  that  that  is  the  limit  beyond  which  the 
human  mind  cannot  go. 

§  2.  If  we  set  out  to  make  an  investigation  concern- 
ing any  subject,  N,  and,  in  the  process  of  our  investiga- 
tion, become  possessed  of  three  judgments,  which  we 
put  forth  in  the  form  of  propositions,  as  follows : 

N  —  d; 
D  -  j; 

J     -X, 

we  may  at  once  apply  to  such  propositions  the  dictum 
of  Aristotle,  by  extending  it,  as  follows — 

I  first  repeat  the  dictum  : 

"Whatever  is  predicated  (/.  e.,  affirmed  or  denied) 
universally,  of  any  class  of  things,  may  be  predicated,  in 
like  manner  {viz.,  affirmed  or  denied),  of  any  thing  com- 
prehended in  that  class." 

As  extended  it  will  read  : 

Whatever  is  predicated  (L  e.,  affirmed  or  denied)  uni- 
versally, of  any  class  of  things,  may  be  predicated,  in 
like  manner  (viz.,  affirmed  or  denied),  of  any  class  com- 
prehended in  that  class;  and,  in  like  manner,  of  any 
thing  comprehended  in  any  class  so  comprehended. 


20  LOGIC   AS   A   PURE   SCIENCE. 

We  have  in  our  last  proposition  predicated  X  (x)  of 
the  whole  class  J,  and  in  the  second  proposition  have 
shown  that  the  class  D  is  comprehended  in  the  class  J. 
X  (x)  may  therefore  be  predicated  of  the  class  D.  But 
we  have  also  show^n  in  the  first  proposition  that  [N"  (which 
may  be  either  a  class,  or  some  single  thing)  is  compre- 
hended in  the  class  D.  We  are  therefore  warranted,  by 
the  dictum  as  extended,  in  predicating  X  (x)  of  N  ;  viz. : 

X    —    X. 

Stating  the  propositions  in  their  reverse  order,  and 
appending  to  them  the  proposition  thus  justified,  with 
the  word  '^therefore"  prefixed,  we  shall  have  the  follow- 
ing expression,  which  is  a  Sorites  ;  viz. : 


J 

— 

x; 

D 

— 

j; 

X 

— 

ci; 

.-.  X 



X. 

But  we  may,  without  reversing  the  order  of  the 
propositions,  append  the  new^  proposition,  and  will  have 
the  same  Sorites,  but  in  a  different  form  ;  viz. : 


X  - 

-  d; 

D  - 

-  J; 

J  - 

-  x; 

X  - 

-    X. 

The  conclusiveness  of  the  reasoning  in  both  forms  is 
apparent. 

§  3.  Thus  we  have  a  complete  Syllogism  (but  in  two 
different  forms  or  figures)  consisting  of  four  propositions, 
composed  of  four  terms. 


SORITES.  21 

Let  lis  now  proceed  to  analyze  it,  and  in  the  course 
of  the  analysis  I  shall  give  new  names  to  the  terms  and 
propositions,  which  mil  be  used  when  referring  to  them 
as  parts  of  the  Sorites,  so  as  to  distinguish  them  from 
like  parts  of  a  simple  Syllogism,  which  will  be  called, 
when  referred  to  as  such,  by  their  old  names. 

And  1st,  as  to  the  terms. 

The  subject,  X,  with  which  we  set  out,  is  equiva- 
lent to  the  minor  term  as  we  have  hitherto  employed 
it.  I  call  it  the  magnus  term  of  the  Sorites,  in  the 
sense  of  holding  a  chief  position  ;  it  being  the  principal 
thing  about  which  we  are  concerned. 

The  two  terms,  D  and  J,  are  each  greater  {major) 
than  the  magnus  term  in  the  forms  above  exhibited 
(which  you  ^vill  hereinafter  lind  are  the  perfect  forms), 
but  one,  D,  is  less  {minor)  than  the  other,  J.  They  are 
both  middle  terms,  and  are  each  once  distributed,  and 
are  compared,  one  with  one  of  the  other  terms,  and  the 
other  with  the  other,  in  the  first  and  third  propositions, 
and  with  each  other  in  the  second.  They  will  be  called, 
D,  the  minor -middle^  and  J,  the  major-middle  terms. 

The  term  X  is  equivalent  to  the  major  term  as 
hitherto  employed,  but  is  greater  than  the  major-middle 
term,  and  is  the  greatest  of  all  the  terms  of  the  Sorites. 
It  will  therefore  be  called  the  maximus  term. 

The  four  terms,  as  in  the  case  of  a  simple  Syllogism, 
occur  twice  each,  the  magnus  and  maximus  terms  each 
once  in  the  premises  (first  three  propositions)  and  once  in 
the  concluding  proposition,  and  the  minor-middle  and 
major-middle  terms  each  twice  in  the  premises. 


22  liOGIC   AS   A   PURE   SCIENCE. 

N  and  X  are  letters  in  the  words  jnagnus  and  maxi- 
TTius  respectively,  and  will  serve  to  keep  their  logical 
significations  in  mind,  in  like  manner  as  the  letters  N, 
D,  and  J,  in  the  words  minor ^  Tniddle,  and  Tnajor^  have 
hitherto  served  in  respect  to  their  logical  significations : 
but  they  will  not  in  their  future  use  so  serve  invari- 
ably. 

2d.  As  to  the  propositions. 

Three  are  premises,  and  will  be  called  from  the  names 
of  the  terms  occurring  in  them  respectively : 

The  magnus  premise ; 

The  middle  premise  (omitting  the  prefixes  minor  and 
major  as  unnecessary,  there  being  no  middle  premise 
in  a  simple  Syllogism) ; 

The  maximus  premise. 

The  concluding  proposition  will  hereinafter  be  found 
to  be  the  ultimate  one  of  two  conclusions  warranted  by 
the  premises ;  and  to  distinguish  it  as  such,  I  shall  call 
it  the  ultima  (conclusio  understood). 

8d.  As  to  the  figure. 

The  figure  of  a  simple  Syllogism  depends  upon  the 
positions  of  its  terms,  but  that  of  a  Sorites  upon  the 
positions  of  its  magnus  and  maximus  premises.  It 
will  be  called  the  configuration.  There  are  two,  the  first 
called  regressive^  in  which  the  maximus  premise  is  the 
first,  and  the  magnus  last ;  and  the  second,  progressive, 
in  Avhich  the  magnus  premise  is  the  first  and  the  maxi- 
mus last.  The  progressive  configuration  was  the  only 
one  known  until  about  the  beginning  of  the  seventeenth 


SORITES.  23 

century,  when  the  regressive  was  discovered  by  a  Ger- 
man logician  named  Goclenius ;  and  it  is  called  also 
Goclenian  after  him.  It  has  been  a  subject  of  dispute 
among  logicians  as  to  which  configuration  should  be 
called  progressive,  and  which  regressive,  but  the  prevail- 
ing opinion  is  in  favor  of  the  names  as  herein  used. 
They  are  generally  treated  of  in  the  order  as  in  the  last 
sentence  ;  but  I  have  reversed  it,  exhibiting  the  regress- 
ive first,  and  the  progressive  last.  The  moods  of  each 
configuration,  and  their  number,  will  hereinafter  appear. 

§  4.  If  all  Sorites,  in  respect  to  the  positions  of  the 
teims,  were  in  the  forms  hereinbefore  given,  and  their 
conclusiveness  were  equally  as  apparent,  I  might  at  once 
proceed  further  to  illustrate  and  comment  upon  them, 
and  state  the  rules  usually  given  in  logical  treatises  con- 
cerning them,  which  are  applicable  only  in  such  case ; 
but  such  is  not  the  case,  and  I  defer  further  comment 
until  I  jshall  have  exhibited  them  in  another  aspect  in 
which  they  can  be  considered  ;  viz.,  as  complex  expres- 
sions consisting  of  two  Enthymemes. 

The  Sorites,  so  to  be  exhibited,  will  be  the  same  as 
before  given ;  and  for  the  sake  of  brevity,  I  shall  call 
the  terms  and  propositions  by  the  names  hereinbefore 
given  to  them,  in  advance  of  exhibiting  them  under  the 
new  aspect. 

§  5.  For  the  purpose  of  such  consideration  I  repeat 
the  three  propositions  with  which  we  set  out. 

X  —  d; 

D-  j; 

J  —  X. 


24  LOGIC   AS  A   PURE   SCIENCE. 

If  now,  having  possessed  ourselves  of  these  judgments, 
but  failing  to  observe,  from  their  perfect  concatenation, 
that  we  may  at  once  deduce  from  them  the  ultimate  con- 
clusion wrapped  up  in  them,  we  proceed  to  syllogize 
with  them  by  means  of  simple  Syllogisms  of  three  prop- 
ositions, we  shall  naturally  commence  with  the  widest 
truth  which  we  have  discovered,  viz.,  J  —  x ;  and  we 
shall  find  our  first  Syllogism  to  be  as  follows  : 

J  —  x; 

D-  j; 
.-.  D  -  X, 

and,  having  thus  become  possessed  of  a  new  truth,  viz.^ 
D  —  X,  we  shall  put  it  forth  as  a  premise,  combining 
with  it  our  first  proposition,  as  yet  unemployed,  and 
produce  a  second  Syllogism  as  follows  : 


D 

— 

x; 

N 

— 

d; 

N 



X. 

The  conclusion  of  this  second  Syllogism  is  the  ultima 
of  the  Sorites,  as  we  have  before  seen  it. 

But  if,  in  the  course  of  our  investigation,  we  had 
stopped  after  the  discovery  of  the  first  two  truths,  viz.  : 

K  —  d; 
D  -  J, 

and  had  syllogized  with  them,  we  should  in  like  manner 


SORfTES.  25 

have  commenced  with  the  widest  truth  then  discovered, 
viz.,  D  —  j,  and  our  first  Syllogism  would  have  been : 

D  -  J: 

N  —  d; 

.••  ^  -  j. 

The  question  would  then  naturally  have  arisen,  But 
what  is  J?  and  resuming  the  process  of  investigation, 
we  should  have  discovered  that  J  —  x,  and  thereupon 
would  have  syllogized  again  : 

J  -x; 

^'-  J; 

.*.    N    —    X, 

and  thus,  by  a  second  series  of  Syllogisms,  we  should 
have  arrived  at  the  ultima  of  the  Sorites,  as  we  have 
before  seen  it. 

By  the  former  process,  we  retraced  our  steps  after 
having  reached  the  summit  of  our  investigation,  and 
it  is  therefore  properly  called  regressive ;  by  the  latter 
w^e  have  reasoned  as  we  progressed,  and  it  is  therefore 
properly  called  progressive ;  but  by  both  processes  we 
have  arrived  at  the  same  ultimate  conclusion,  illustrat- 
ing the  axjhorism  that  ''all  truth  is  one." 

The  middle  premise,  as  you  will  observe,  is  the  minor 
premise  of  the  first  Syllogism  in  the  first  series,  and  the 
maxim  us  premise  the  major ;  and  the  middle  premise 
is  the  major  premise  of  the  first  Syllogism  in  the  second 
series,  and  the  magnus  premise  the  minor ;  and  all  the 


26  LOGIC   AS   A   PURE   SCIENCE. 

Syllogisms  are  in  Barbara  in  the  first  figure,  which  you 
have  learned  is  the  only  j)erfect  figure. 

§  6.  But  we  may  reason  imperfectly,  and  that  too, 
even  when  we  have  our  judgments  in  a  perfect  concate- 
nation, as  they  have  thus  far  been  exhibited  ;  and,  in 
such  case,  we  shall  find  our  Syllogisms  to  be  in  one  or 
more  of  the  imperfect  figures.  If,  in  the  regressive 
process  we  begin  to  syllogize  with  the  middle  premise 
as  the  major  premise  of  the  first  Syllogism  (instead  of 
the  minor),  and  the  maximus  as  the  minor  (instead  of 
the  major) ;  and  in  the  progressive  process,  wdth  the 
middle  premise  as  the  minor  premise  of  the  first  Syllo- 
gism (instead  of  the  major),  and  the  magnus  premise  as 
the  major  (instead  of  the  minor),  we  can  frame,  or 
attempt  to  frame,  two  other  series  of  Syllogisms,  which  I 
here  exhibit,  with  the  two  Syllogisms  of  each  series, 
side  by  side,  as  follows : 

In  fhe  regressive  process. 

D  —  j;  X  —  d; 

J   —  x;  y^  N  —  d; 

.-.  X   —  d.  — -^ 

In  the  progressive  process. 

N  —  d;  J   —  x; 

D  —  j ;  ^ J    —  1^  ' 

.-.  j    —  n.  — —^^^^  .*.  11   —  X. 

In  the  latter  series,  only  a  particular  ultimate  con- 
clusion is  arrived  at ;  in  the  former,  no  ultimate  conclu- 
sion is  warranted  by  reason  of  non-distribution  of  the 
middle  term  in  the  second  attempted  Syllogism. 


SORITES.  27 

Thus,  as  you  will  perceive,  imperfect  processes  are 
followed  by  imperfect  or  no  results. 

§  7.  To  recur  now  to  the  two  principal  series,  and  for 
the  purpose  of  bringing  the  two  Syllogisms  of  each 
together,  in  such  a  method  of  arrangement  that  you  may 
at  once  see  the  connection  between  them,  and  the  appli- 
cation of  the  remarks  that  are  to  follow,  I  repeat  them, 
putting  the  two  Syllogisms  of  each,  side  by  side. 

First,  or  regressive  series. 

J  —  x;  ^^ ^D  —  x; 

D-j;  /                N-d; 
.-.  D  — 

Second,  or  progressive  series. 

D-j;  J  -  x; 

X-d;  ^ N-j; 

.-.  X  -  j. -^  .-.  X  -  X. 

By  taking  an  Enthymeme  of  the  third  order  from  the 
first  Syllogism,  and  one  of  the  first  order  from  the  second 
Syllogism  of  the  first  series,  and  putting  them  together 
in  one  expression,  and,  by  taking  an  Enthymeme  of 
the  third  order  from  the  first  Syllogism  of  the  second 
series,  hut  transposing  the  propositions  so  taken^  and 
one  of  the  second  order  from  the  second  Syllogism  of 
the  same  series,  and  putting  them  together  in  one  ex- 
pression, we  shall  have  the  same  Sorites,  as  before,  in 
the  two  configurations,  viz.: 


28  LOGIC   AS   A   PURE   SCIENCE. 

Regressive  Sorites  Proyressive  Sorites 

from  the  first  series.  from  the  second  series. 

J  —  x;  X  —  d; 

D-  j;  D  -  j; 

X  —  d;  J  —  x; 

.-.  X  —  X.  .-.  N  —  X. 

The  conclusion  of  the  first  Syllogism  in  each  series  is 
held  in  the  mind  (otherwise  there  were  no  Enthymeme), 
but  carried  forward  mentally,  and  employed  as  a  pre- 
mise, still  unexpressed,  in  connection  with  the  P]nthy- 
meme  taken  from  the  second. 

A  Sorites  considered  as  a  complex  expression  as 
above  shown  is  also  called  a  Chain-Syllogism. 

§  8.  The  middle  premise  (being  the  proposition  B — j 
in  which  the  minor-middle  and  major-middle  terms  are 
compared)  will  always  be  the  second  proposition  in  every 
Sorites,  simple  (as  hitherto  shown)  or  compound  (as  to 
which  latter  you  will  hereinafter  be  instructed) ;  and  by 
expressing  it,  in  connection  with  the  ultima,  every 
Sorites  may  be  still  further  abridged,  thus  : 

D-  j; 

.-.  N  —  X. 

All  the  four  terms  here  appear,  but  each  only  once. 
Such  an  expression  is  in  the  form  of  an  Entliymeme 
(but  is  not  an  Enthymeme,  for  that  can  have  only  three 
terms),  and  may  properly  be  called  an  Abridged  Sorites. 

From  the  employment  of  the  middle  premise  as  the 
minor  or  major  premise  of  the  first  Syllogism,  I  desig- 
nate Sorites  (considered  as  complex  expressions)  minor 


SORITES. 


29 


and  major  Sorites,  respectively,  for  the  purpose  of  classi- 
fication as  hereinafter  shown.  Either  may  be  regressive 
or  progressive  ;  but  we  shall  see  that  the  proper  division 
of  Sorites  is  into  regressives  and  progressives. 

Observe,  that  in  all  major  Sorites,  but  in  no  minors, 
the  premises  constituting  the  Enihymeme  of  the  third 
order  taken  from  the  first  Syllogism,  are  transposed. 

§  9.  The  Syllogisms  of  the  two  principal  series  (of 
Enthymeines  of  which  the  Sorites  exhibited  consist) 
are  Avholly  in  the  first  figure.  But  a  little  reflection 
will  show  that  Sorites  may  also  consist  of  Enthymemes 
taken  from  Syllogisms  in  any  of  the  figures  capable  of 
combination  in  series,  quantity  and  quality  considered. 
And,  as  all  Sorites  may  be  abridged  in  the  manner 
hereinbefore  shown,  it  is  also  manifest  that  the  range  of 
possible  abridged  Sorites  is  limited  to  the  number  of 
possible  combinations  of  two  proj)ositions  composed  of 
four  terms,  expressed  in  the  same  form  as  to  the  order 
of  the  temis  throughout,  but  modified  in  respect  to 
quantity  and  quality,  as  in  the  following  scheme ;  and 
only  such  can  be  valid  as  are  capable  of  being  expanded 
into  full  Sorites,  and  from  full  Sorites  into  at  least  two 
series  of  Syllogisms.  The  propositions  must  be  in  one  or 
another  of  the  combinations  shown  by  full  lines  in  the 
scheme. 

D   —  i ;  -^:::i ^rr-^^  X  —  x. 


X. 


11    —   X. 


X. 


30  LOGIC    AS   A   PURE   SCIENCE. 

Considering  the  lines  connecting  the  propositions, 
each  as  signifying  ''and  therefore,"  there  are  sixteen  dif- 
ferent combinations.  But  of  these,  only  nine  will  be  found 
to  be  valid,  and  they  are  symbolized  by  the  same  symbols 
as  those  of  valid  Enthymemes  of  the  first  order,  as  here- 
inbefore shown,  and  may  be  expanded  into  full  Sorites 
(the  supplied  premises  varying  in  the  order  of  the  terms 
as  well  as  in  quantity  and  quality),  and  from  full  Sorites 
into  two,  three,  or  four  series  of  Syllogisms,  with  the 
middle  premise  as  either  the  minor  or  the  major  premise 
of  the  first  Syllogism  of  one  or  more  series,  excei^t  in  two 
cases,  which  will  be  hereinafter  noted. 

The  number  of  valid  full  Sorites  into  which  the  nine 
abridged  forms  may  be  so  expanded  is  one  hundred  and 
forty-four,  of  which  one  half  are  minors  and  one  half 
majors,  classified  as  such  according  to  the  combinations 
of  the  symbols  of  the  abridged  forms,  as  follows  : 


Symbols. 

Mi7iors. 

Majors, 

A,  A. 

1 

1 

A,  E. 

4 

8 

A,  I. 

16 

10 

A,  0. 

24 

24 

E,  E, 

4 

4 

E,  0. 

10 

16 

T,   I. 

4 

4 

I,  0. 

8 

4 

0,  0. 

1 

1 

72  72 

The  following  synopsis  exhibits  all  possible  valid  cate- 
gorical Sorites,  in  their  abridged  forms,  as  minors  on  the 


SORITES.  31 

left-hand  pages,  and  as  majors  on  the  right ;  together  with 
the  premises  by  which  they  may  be  expanded  into  valid 
full  Sorites,  and  the  names  of  the  moods  in  which  they 
can  be  further  and  fully  expanded  into  series  of  Syllo- 
gisms. They  are  arranged  in  the  order  A,  I,  E,  O,  of 
the  symbols  of  the  ultima. 

The  abridged  forms  may  be  expanded  into  full  Sorites 
by  writing  first,  the  first  of  the  two  supplied  premises ; 
secondly,  the  middle  premise;  thirdly,  the  second  of  the 
two  supplied  premises  ;  and  lastly,  the  ultima. 

Preceding  the  synopsis  are  given  two  series  of  schemes, 
by  which  the  different  ways  in  which  abridged  Sorites 
may  be  expanded  into  full  Sorites,  and  from  full  Sorites 
into  series  of  Syllogisms,  in  all  combinations  of  figures 
in  which  they  are  capable  of  being  so  expanded,  may  be 
clearly  seen.  The  terms  of  the  abridged  Sorites  are  in 
capitals  enclosed  in  circles  connected  by  lines  represent- 
ing the  copulas  of  the  propositions.  The  curved  lines 
(considered  as  copulas)  above  the  propositions  constitut- 
ing the  abridged  Sorites,  in  connection  A\ith  those  propo- 
sitions, indicate  two  expanded  Sorites,  and  in  connection 
also  with  the  dotted  straight  line  above,  indicate  two 
series  of  Syllogisms ;  and  the  lines  below,  two  other 
expanded  Sorites,  and  two  other  series  of  Syllogisms. 
Tlie  dotted  straight  lines  show  the  unexj)ressed  conclu- 
sions of  the  first  Syllogisms,  which  in  each  case  becomes 
one  of  the  premises  of  the  second.  The  modifications  of 
the  propositions  of  the  abridged  Sorites,  in  respect  to 
quantity  and  quality,  are  indicated  by  the  symbols  above 
and  below  the  lines  representing  their  copulas  respect- 
ively ;    those  above  referring  to  the  Sorites  and  Syllo- 


32  LOGIC   AS   A  PUEE   SCIENCE. 

gisms  indicated  above,  and  those  below,  to  those  below. 
The  modifications  of  the  other  indicated  propositions 
are  also  in  like  manner  signified. 

The  symbols  in  connection  with  the  lines  are  those 
only  in  which  the  Sorites  and  Syllogisms  are  valid  in  the 
figures  shown. 

It  is  not  meant  that  each  symbol  in  connection  with 
each  other  will  yield  a  valid  Sorites,  but  that  each,  in 
<;onnection  with  some  one  or  more  of  the  others,  will 
be  found  valid.  Thus,  in  the  second  scheme  of  minors, 
the  maximus  premise,  A,  will  combine  with  the  middle 
premise  as  E  or  O,  and  E  with  A  or  I,  but  not  other- 
wise. 

The  designations  of  premises,  written  between  paral- 
lel curved  lines,  refer  to  the  propositions  indicated  by 
both  lines ;  the  symbols  and  number  of  the  figure  being 
on  the  other  side  of  each  line,  respectively. 

By  marking  all  the  lines  with  all  the  symbols,  you 
will  be  able  to  make  an  exhaustive  analysis  of  all  possi- 
ble ways  in  which  it  may  be  attempted  to  frame  simple 
Sorites.  In  view  of  the  number  given  on  the  next  page, 
you  may  think  the  attempt  formidable,  but  you  will  find 
it  not  so  much  so  as  it  will  at  first  appear,  if  you  but 
consider  and  apply  to  the  symbols  the  rules  of  the  syllo- 
gism before  proceeding  to  test  them.  The  lines  above  the 
proi3ositions  constituting  the  abridged  Sorites  are  marked 
with  all  the  symbols  of  the  propositions  respectively,  as 
they  may  be  employed  in  single  simple  syllogisms,  as 
hereinbefore  shown,  but  those  below,  not ;  and  if  you 
first  add  to  the  latter  the  omitted  symbols,  making  them 
to  correspond  with  those  above,  you  will  find  that  such 


SORITES.  33 

added  symbols  will,  in  all  cases,  yield  no  conclusion  in 
the  second  of  the  Syllogisms,  by  reason  of  one  or  the 
other  of  the  two  faults,  undistributed  middle  and  illicit 
process  of  the  major.  If  the  remaining  symbols  be  then 
added  to  each  line,  a  violation  of  some  one  or  more  of 
the  rules  of  the  syllogism  will  be  found  in  either  the 
first  or  second  Syllogism. 

The  total  number  of  the  ways  in  which  it  may  thus 
be  attempted  to  combine  the  four  symbols  A,  E,  I,  O. 
according  to  the  schemes  is  eight  thousand  one  hundred 
and  ninety-two,  that  being  the  product  of  the  number  of 
ways  (256)  in  which  the  four  symbols  may  be  combined 
(all  the  same,  or  partly  the  same,  or  all  different),  multi- 
plied by  the  number  of  combinations  of  propositions  (4) 
indicated  by  each  scheme,  and  again  by  the  number  of 
schemes  (8)— (256  x  4  x  8  =  8192).      . 

The  total  number  of  valid  Sorites  without  regard  to 
their  character  as  minor  or  major,  or  as  regressive  or  pro- 
gressive, will  be  hereinafter  found  to  be  forty-four. 

By  examining  each  scheme,  and  comparing  the 
Sorites  and  series  of  Syllogisms  thereby  indicated  (those 
above  with  each  other,  and  those  below  with  each  other), 
and  by  comparing  each  scheme  with  each  of  the  others 
in  all  possible  ways,  the  differences  between,  and  con-e- 
lations  of,  the  several  figures  of  the  Syllogism  and 
the  two  kinds  of  Sorites  indicated  by  the  schemes  (that 
is,  either  minor  or  major),  will  also  clearly  appear,  and 
the  student  cannot  fail  to  be  impressed  with  the  har- 
mony and  symmetry  of  pure  reasoning,  in  all  its  varied 
possible  forms  of  expression. 


34 


LOGIC   AS   A   PURE   SCIENCE. 


SCHEMES    OF    MINOR    SORITES. 

FIRST   SYLLOGISM    IN    FIRST    FIGURE. 

A  J^E .  or  O. ^ :^ 

1    __- isUig,^ ^^^^iPren^   . 

4.  J  p,  A.  or  E.  _J^^^\V^ 

^^.  I.  or  E. 

FIRST   SYLLOGISM    IN    SECOND    FIGURE. 

]5-_or_(). i. 

@4^M^0  •■•  0-1^© 

.      ^                                A.  orE.                     StA3^^'"^VN^ 
^      -f^> __^-— ^^^^ 


SORITES. 


35 


SCHEMES    OF    MAJOR    SORITES. 


FIRST   SYLLOGISM    IN    FIRST    FIGURE. 


A  J:-5-_"J-_0: 

Min.  prerti.        A^  o^  E. 


1st 


A.  or  I 


A.L  or  £. 


FIRST   SYLLOGISM    IN   SECOND   FIGURE. 

E.  or  O. 


Jst  fig. 


E. 


LOGIC    AS   A  PURE   SCIE]S^CE. 


SCHEMES    OF    MINOK    SORITES. 

FIRST   SYLLOGISM    IN    THIRD    FIGURE. 

I-P£9-_ 


FIRST   SYLLOGISM   IN   FOURTH   FIGURE. 

I.  E.  or  O. 


o^T 


A.o^ 


I.  or  E. 


SORITES. 


37 


SCHEMES    OF    MAJOR    SORITES. 

FIRST   SYLLOGISM    IN    THIRD   FIGURE. 

f 

^ J'_orJl- 

Min.  prem. 

^^^^.  ^l: -^^^    ^ret^^- 

A.  or  I. 

i: 

FIRST   SYLLOGISM   IN    FOURTH   FIGURE. 

^  OP  THE    "nP^^ 

1.^-0, ffUH I V E E SI T 1 

r ^!^^ 

-^^  orT    - — ^*^=^ — i 

A.  or  E. 
______ 

1 


LOGIC  AS  A  PURE   SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms  of 

TOGETHER  WITH  ALL  THEIR  POSSIBLE  MAGNUS  AND  MAXIMUS 
PREMISES, 


Sym- 

BOLS. 

Nos. 

Abridged  Minor 

Sorites. 

Middle  Premise,   Ultima. 

Mood  of 

First 
Syllogism. 

Major  Premise 

OF  First 

Syllogism. 

Minor  or 

Major  Premise 

of  Second 

Syllogism. 

Mood  of 

Second 

Syllogism. 

A,  A. 

1 

D-j;   /.  N-x. 

bArbAra 

J-x 

N-d 

barbArA 

A,  I. 

2 

D— j ;   .*.   n  —  X. 

bArbAra 

J-x 

N-d 

a,AJ.  1st  fig. 

3 

//                 /' 

>i 

II 

n-d 

darll 

4 

//                 It 

\  or,  A,  A,  I     ■ 

ri 

D-n 

J  darAptI 
\  disAmls 

5 

rt                          If 

bArbAra 

II 

d-n 

datlsl 

6 

II                          II 

brAmAntip 

N-d 

J-x 

dAtisI 

7 

II                          If 

dlmAris 

n-d 

II 

It 

8 

II                          II 

dArApti 

D-n 

II 

It 

9 

II                           II 

dIsAmis 

d-n 

If 

It 

10 

II                          II 

J  bArbAra      1 
1  or,  A,A,i     \ 

J-n 

D-x 

J  dAraptI 
1  dAtisI 

11 

II                       n 

bArbAra 

1' 

d-x 

dlsamls 

12 

II                       It 

II 

If 

X-d 

brAmarUIp 

13 

II                       ti 

II 

II 

x-d 

dimarls 

14 

II                       II 

dArApti 

D-x 

J-n 

disAmIs 

15 

If                       If 

dIsAmis 

d-x 

// 

It 

16 

11                       If 

brAmAntip 

X-d 

// 

II 

17 

II                       If 

dImAris 

x-d 

// 

II 

1,1. 

18 

d-j;   .-.    n-x. 

dArIi 

J-x 

D-n 

disAmIs 

19 

fi                 II 

dAtlsi 

D-n 

J-x 

dAtisI 

20 

If                 It 

dArn 

J-n 

D-x 

If 

21 

II                 II 

dAtlsi 

D-x 

J— n 

disAmIs 

SORITES. 


39 


Abridged  Categorical  Sorites. 


AND  MOODS  OF  SIMPLE  SYLLOGISMS  IN  WHICH  THEY  CAN  BE  FULLY 
EXPANDED. 


Sym- 
bols. 

N08. 

Abridged  Major 

Sorites. 

Middle  Premise.   Ultima. 

Mood  of 

First 
Stllogism. 

Major  or 

Minor  Premise 

of  Seookd 

Syllogism. 

Mood  of 

Second 

Syllogism. 

1 

A,  A. 

D-j;   .-.  N-x. 

bArbAra 

N-d 

J-X 

/jArbarA 

A,  I. 

2 

D — j ;   /.    n  — X. 

j  bArbAra      1 
1or,  ^,^,1    f 

N'-d 

J-X 

SA.a.I 
}  dAril 

3 

tt                It 

dArIi 

n-d 

" 

f 

^ 

ir                         It 

dAvApti 

D-n 

It 

5 

n                          It 

dAiM 

d-n 

ft 

It 

6 

tt                          II 

brAmAntip 

J-x 

D-n 

dimArls 

7 

ri                             II 

dArApti 

D-x 

J-n 

» 

8 

ft                             II 

dAtM 

d-x 

tr 

II 

9 

ti                             II 

J  bArbAra       I 
"1  or,  A,A,i    S 

X-d 

If 

j  braiPAntIp 
1  dimArls 

10 

It                             II 

dArIi 

x-d 

It 

11 

11 

tt                             II 

brAmAntip 

J-n 

D-X 

dAHI 

1,1. 

12 

d— j  ;   .*.    n— X. 

dIsAmis 

D^n 

J-x 

dAril 

13 

//                 It 

dImAris 

J-x 

D-n 

dimArls 

14 

II                 It 

dIsAmis 

D-x 

J-n 

It 

15 

It                 tt 

dImAris 

J-n 

D-x 

dAHI 

40 


LOGIC  AS   A   PURE  SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms  of 

TOGETHER  WITH  ALL  THEIR  POSSIBLE  MAGNUS  AND  MAXIMUS 
PREMISES, 


Sym- 
bols. 

Nos. 

Abridged  Minor 

Sorites. 

Middle  Premise,   Ultima. 

Mood  of 

First 

Syllogism. 

10 

III 

Minor  or 

Major  Premise 

OF  Second 

Syllogism. 

Mood  of 

Second 

Syllogism. 

A,  E. 

22 

D-j  ;   .-.  S-X. 

cElArent 

^-X 

N-d 

celArEnt 

23 

//                        n 

cEsAre 

X-J 

// 

" 

24 

ti                        11 

cElArent 

d^-N 

X-d 

cAmenEa 

25 

n                       II 

cEsAre 

}^-J 

II 

II 

E,  E. 

26 

B-_J;  ...  5?_x. 

cAmEstres 

x-j 

N-d 

cdArEnt 

27 

II                          II 

cAmEnes 

N-d 

x-j 

cAmenEs 

28 

II                      II 

II 

X-d 

N-j 

CdArEnt 

29 

II                          II 

cAmEstres 

N-J 

x-d 

cAmenEs 

A,  0. 

30 

D-j;   .-.  n-^X. 

cElArent 

dF-X 

N-d 

j  e,  A,  0. 
\           Ist  fig. 

31 

//                          n 

II 

II 

n-d 

ferlO 

32 

II                          II 

II 
■  or,  E,  A,o\ 

1' 

D  — n 

SfelAptOn 
\  hoklrdO 

33 

n                       II 

cElArent 

II 

d-n 

ferlsO 

34 

It                       II 

cEsAre 

X-J 

X-d 

J  e,  A,  0. 
\          l.st  fig. 

35 

II                       II 

n 

II 

n-d 

ferlO 

36 

II                       II 

\  or,  E,  A,o] 

II 

D-n 

jfef,Aj)tOn 
1  bokArdO 

37 

II                       It 

cEsAre 

II 

d-n 

ferlsO 

38 

II                       II 

cElArent 

dF-N 

X-d 

j  A,  e,  0. 
\         4rh  fig. 

39 

II                       II 

cEsAre 

5?-J 

" 

// 

SORITES. 


41 


Abridged  Categorical  Sorites.     (Continued.) 

AND  MOODS  OF  SIMPLE  SYLLOGISMS  IN  WHICH  THEY  CAN  BE  FULLY 

EXPANDED. 


Sym- 
bols. 

Nos. 

Abridged  Major 

Sorites. 

Middle  Premise.  Ultima, 

Mood  op 
First 

S-XXLOGIBM. 

1   . 
Ill 
PI 

Major  or 

Minor  Premise 

OP  Second 

Stllogism. 

Mood  of 

Second 

Syllogism. 

A,E. 

16 

D-j; 

.-.  J^-X. 

bArbAra 

N-d 

J_X 

cElarEnt 

17 

n 

ft 

" 

It 

X  — J  !    cEearE 

18 

It 

II 

" 

X-d 

J  —  X     camEnEs 

19 

ti 

It 

II 

It 

N-J 

camEstrEa 

20 

It 

It 

cAmEnea 

J-X 

N-d 

cesArE 

21 

ft 

tt 

cAmEstres 

X-J 

II 

It 

22 

It 

tt 

cAmEnes 

J-N 

X-d 

cAmestrEg . 

23 

II 

II 

cAmEstres 

}?-J 

// 

II 

E,  E. 

24 

B-J; 

.-.  ^-X. 

cElArent 

N-d 

X-j 

cAmestrEs 

25 

II 

II 

cEsAre 

X-j 

N-d 

cesArE 

26 

It 

If 

n 

N-j 

X-d 

cAmestrEg 

27 

II 

It 

cElArent 

X-d 

N-j 

cesArE 

A,  0. 

28 

D-j; 

.-.  n-.^X. 

S  bArbAra    \ 
1  or,  A,A,i\ 

N-d 

J-X 

\f^S 

29 

'/ 

;/ 

dArR 

n-d 

If 

It 

30 

/; 

// 

dAvApti 

D-n 

" 

ft 

31 

II 

tt 

dAtlsi 

d-n 

II 

tl 

32 

II 

It 

^  bArbAra    1 
1  or,  A,  A,  is 

N-d 

X-J 

\Ea.  0 
IfEstinO 

33 

It 

It 

dAvR 

n-d 

" 

tt 

34 

II 

II 

dArApti 

D-n 

// 

It 

35 

tt 

II 

dAtM 

d-n 

// 

It 

36 

It 

It 

bArbAra 

X-d 

J^-N 

J  a,  E,  0. 

\           4th  fig. 

37 

II 

tt 

It 

It 

N-J 

j  o,  E,  0. 

1             2d  fig. 

38 

tl 

tt 

It 

II 

n-«-,j'    barOkO 

1 

42 


LOGIC   AS   A   PURE   SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms  of 


TOGETHER  WITH  ALL  THEIR  POSSIBLE  MAGNUS 
PREMISES, 

AND  MAXIMUS 

1 

a 

1      w 

Sym- 
bols. 

N08. 

Abridged  Minor 

Sorites. 

Middle  Premise.   Ultima. 

Mood  of 

First 

Syllogism. 

Minor  or 

Major  Premie 

of  Second 

Syllogism. 

Mood  of 

Second 

Syllogism. 

A,  0. 

40 
41 

I)_j;     ...    ll__X. 

n                              tt 

fElAptm 
hOkArdo 

B^-X 

d-^X 

J-n 

bokArdO 

II 

42 

It                              It 

fEsApo 

5-D 

It 

II 

43 

It                              It 

\hArhAra    \ 
1  or,  A,AA\ 

J-n 

D-X 

SfElapt  On 
TfErisO 

44 

II                              It 

hArhAra 

// 

d-wX 

bOkardO 

45 

II                              II 

II 
\  or,  A,  A,i 

It 

X-D 

SfEsapO 
ifrEsisOn 

46 

II                              II 

hrAniAntip 

N-d 

J-X 

fErisO 

47 

II                           r 

dlniAris 

n-d 

ti 

11 

48 

It                           1' 

dArApti 

D~n 

II 

49 

11                           r 

dIsAmis 

d-n 

1' 

" 

50 

II                           1' 

brAmAntip 

K-d 

X-J 

frEsisOn 

51 

II                           n 

dImAris 

n-d 

/' 

II 

52 

II                           It 

dArApti 

D-n 

I' 

II 

53 

It                           " 

dIsAmis 

d-n 

II 

I,     0. 

54 

d-j;    .-.  11  ^X. 

/Brio 

^-X 

D-n 

bokArdO 

55 

II                   It 

fEstIno 

X-J'      " 

It 

56 

II                   It 

fErlso 

^-X\  J-n 

" 

57 

'1                   II 

frEsIson 

X-D 

'/ 

It 

58 

II                   II 

dAtlsi 

D-n 

^-X 

fErisO 

59 

It                   It 

II 

It 

X-J 

frEmOn 

60 

n                      II 

dArIi 

J-n 

D-X 

fErisO 

61 

II                      II 

II 

X-D 

frEsisOn 

SORrtES. 


43 


Abridged  Categorical  Sorites.     (Continued.) 


AND 

MOODS  OF  SIMPLE  SYLLOGISMS   IX    WHICH   THEY  CAN 

BE  FULLY 

EXPANDED. 

Sym- 

BOLS. 

Nos. 
30 

Abridged  Major 

Sorites. 

Middle  Premise.   Ultima. 

Mood  of 

First 
Syllogism. 

Minor  Premise 

OF  First 

Syllogism. 

Major  on 

Minor  Premise 

OF  Second 

Syllogism. 

Mood  of 

Second 

Syllogism. 

A,  0. 

I)_j:    ...  n-^X. 

IrAmAntip 

J — 11 

B-X 

fEriO 

40 

'/                       It 

II 

11 

X-l) 

fmtinO 

41 

II                       II 

cAmEnes 

^-X 

■vr       .1    j  6,  A,  0. 
^—^^\            2dfig. 

42 

II                       II 

'■ 

'■ 

n  — d      festlnO 

43 

It                       II 

•' 

" 

D  —  n      fesApO 

44 

It                       f 

II 

d  —  n      fresh  On 

45 

ir                               It 

cAmEstref! 

X-J 

if-cij''^-?i«g. 

46 

It                               If 

It 

n  — d      festInO 

47 

It                               1' 

" 

II 

D  — 11      fesApO 

48 

II                               II 

II 

II 

d-n 

fresIsOn 

49 

II                              t' 

icAmEnes    \ 
j  or,  A,E,os 

^-N 

X-d 

SA,e,0 
UArokO 

50 

II                               '1 

J  cAmEstres  \ 
\  or,  A,E,oS 

?^-J 

ti 

\A,e,0 
}  bArokO 

51 

II                               II 

bArOko 

n^J 

II 

It 

I.   0. 

52 

d  — j;    .-.  n-wX.    disAmis 

D-n 

J-X 

fEriO 

53 

i         " 

/' 

X-J 

fEstinO 

54 

II                            n 

dImAris 

J-ii 

&-X 

fEnO 

55 

It                            It 

It 

'' 

X-D 

fEsHnO 

44 


LOGIC   AS  A   PURE   SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms  of 

TOGETHER  WITH  ALL  THEIR  POSSIBLE  MAGNUS  AND  MAXIMUS 
PREMISES, 


Sym- 


Nos. 


E,  0. 


0,  0. 


62 

63 
64 
65 

66 

67 

68 

69 
70 


71 


Abkidged  Minor 

Sorites. 

Middle  Premise.  Ultima. 


^- J  ;  .-.   ii-wX. 


Mood  op 

First 
Syllogism. 


«  £  £ 
Kg  3 


cAmEstres 


or,  A,  E,  o 
cAmEstres 

cAmEnes 


x-j 


X-d 


I  or,  A,E,o    I  I 
cAmEnes  " 


N-J 


72     d-^J  ;  .'.   n — 'X.    bArOko 


x-j 


§fi«g 


X-d 

n-d 
D-n 
d-n 

N-j 

J-n 
X-j 


X-d 


Mood  op 

Second 

Syllogism. 


e,A,  0. 1st  fig. 

ferlO 

SfdAptOn 
I  boklrdO 

ferlsO 
^,^,aistfijr. 

ferlO 

SfdAptOn 
1  bokArdO 

ferlsO 

A,e,0.  ith&g. 


^,f,^.4tlil 


D  —  n  I    bokArdO 


SOKITES. 


45 


Abridged  Categorical  Sorites.    (Concluded.) 

AND  MOODS  OF  SIMPLE  SYLLOGISMS  IN  WHICH  THEY  CAN  BE  FULLY 
EXPANDED. 


Sym- 
bols. 

N08. 

56 

Abridged  Major 

Sorites. 

Middle  Premise.   Ultima. 

Mood  of 

First 
Syllogisic 

H 

mi 

Mood  of 

Second 

Syllogisji. 

E,  0. 

^_J;...    n-^X. 

1  or,  E,A,o\ 

N-d 

x-J 

SA,€,0 
bArokO 

57 

'r                           /' 

fErlo 

n-d 

'' 

It 

58 

f                       n 

fETApton 

D-n 

II 

II 

59 

"                       II 

fErlso 

d-n 

" 

It 

60 

tt                       It 

j  cEsAre        \ 
\  or,  E,A,ofi 

^-] 

x-d 

iA,e,0 
\  hArokO 

61 

n                         II 

fEslIno 

n-j 

II 

It 

62 

II                         II 

fEsApo 

J-n 

" 

11 

63 

It                         It 

frEsIaon 

j-n 

II 

" 

64 

'1                         II 

cEsAre 

X-j 

N  — d      ^,4,0.2dfig. 

6b 

'1                         II 

II 

It 

n-d 

featlnO 

66 

n                          II 

II 

II 

D-n 

fesApO 

67 

II                          1' 

If 

It 

d-n 

fresh  On 

68 

It                          II 

cElArerU 

X-d 

N-j 

e.A.aadfig. 

69 

II                          II 

II 

It 

n-j 

fegtInO 

70 

II                          II 

II 

II 

J-n 

fesApO 

71 

If                          II 

It 

It 

j-n 

f res  Is  On 

0,  0.|72 

d — ^J  ;  .*.    n — 'X.     hOkArdo 

D-n 

X-j 

bArokO 

46 


LOGIC   AS  A   PURE   SCIENCE. 


§  10.  By  examining  the  foregoing  synopsis  and  testing 
the  same,  it  will  be  found  that 

If  the  major-middle  term  (predicate  of  the  middle  premise)  he 
the  middle  term  of  the  first  Syllogism,  the7i  if  the  Sorites  he 


Minor  : 


hut  if  it  be 


and  the  configurations  of 
the  Sorites,  whether  re- 
gressive or  progressive,  or 
bofn,  and  the  number  of 
each,  will  be  as  follows : 


Reg.,  G 

Reg.,i  6 

Reg.,  6 

3    I  Reg.,'  6 


Prog., 

1 
6 

2 

Prog., 

6 

2 
-4 

Prog., 

4 

J, 

Major  ; 


and  the  configurations  of 
the  Sorites,  xchethe)-  re- 
aressive  or  progressive,  or 
both,  and  the  number  of 
each,  will  be  as  follows : 


2  Reg., 

4  Reg., 

2  Reg., 

Jf  I  Reg., 


0   Prog.,     G 


4; 

Prog., 
3   Prog., 

4 


But  if  the  minor-middle  term  (suhject  of  the  middle  pj^emise) 
he  the  middle  term  of  the  first  Syllogism,  then,  if  as  sec- 
ondly ahove,  the  figures  of  the  Syllogisms  may  he,  and  the 
configurations  of  the  Sorites  and  the  numher  of  each  ivill  he 
as  follotvs : 


Minor ; 

Major  ; 

1 

1 

Prog., 

6 

3 

3 

Reg., 

G 

Prog., 

G 

1 

2 

Reg., 

6 

Prog., 

6 

3 

-^ 

Prog., 

3 

1 

J^ 

Reg., 

G 

4 

1 

Reg., 

3 

3 

1 

Prog., 

6 

Jf 

r, 

Reg., 

6 

Prog., 

4 

3 

2 

Prog., 

G 

Jf 

u 

39 

Prog., 

4 
33 

3 

Jf 

Reg., 

3 
32 

40 

Total  minors,  72  ;  Total  majors,  72  ; 

Grand  total,  144. 


SORITES.  47 

But,  by  a  careful  examination  of  the  synopsis,  it  will 
be  found  that  fifty-six  of  the  Sorites  are  both  minors 
and  majors.  That  number  must  therefore  be  deducted 
from  the  grand  total,  leaving  eighty-eight  different 
forms. 

Each  of  the  four  figures  occurs  as  the  figure  of  the 
first  Syllogism  in  both  minor  and  major  Sorites  ;  but  the 
second  does  not  occur  as  the  figure  of  the  second  Syllo- 
gism in  minors,  nor  the  third  in  majors.  With  these 
exceptions,  all  the  figures  occur  also  as  figures  of  the 
second  Syllogism. 

The  following  is  a  synopsis  of  all  the  eighty-eight 
possible  forms  of  valid  simple  Sorites  arranged  according 
to  their  configurations,  regressives  on  the  left-hand  pages, 
and  progressives  on  the  right,  and  without  regard  to  their 
being  either  minor  or  major,  but  showing  in  the  columns 
on  the  left-hand  side  of  each  page,  the  moods  of  the 
Syllogisms  in  respect  to  which  they  are  minors,  and  on 
the  right,  those  in  respect  to  which  they  are  majors. 

There  will  be  found  on  the  pages  of  regressives,  seven- 
teen, and  on  the  pages  of  progressives,  fifteen,  in  Avhich 
the  moods  are  only  on  one  side,  leaving  twenty-seven 
regressives  and  twenty-nine  progressives  in  which  the 
moods  are  on  both  sides,  and  which  together  make  the 
fifty-six  alike  on  both  sides  of  the  preceding  synopsis, 
as  above  stated.  Two,  namely,  Xos.  25  and  38,  are  the 
exceptions  hereinbefore  referred  to.  No.  25  is  a  minor 
Sorites  only,  and  No.  38  a  major  Sorites  only,  in  both 
configurations. 

As  before,  they  are  arranged  in  the  order  A,  I,  E,  O 
of  the  symbols  of  the  ultima. 


48 


LOGIC   AS   A  PURE   SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms 


Series  of  Syllogisms 

IN  WHICH  THE  MIDDLE 

If 

Regressive  Configuration. 

Series  of  Syllogisms 
in  which  the  middle 

Premise 

IS    Minor, 

Maximus 

Major    of 

o 

1 

1 

Premise    is     Major, 

AND       THE 

Premise 

THE  riBST 

Maximus    Middle 
Premise.   Premise. 

Magnus 
Premise. 

1 
Ultima. 

AND    THE    Maximus 
Premise    Minor    of 

THE  FIRST. 

bArbAra 

barbArA 

J-X 

D-j 

X-d 

/.X"— x 

bArbAra 

,a,A,I.  1 

1  1st  fig.  r 

2 

J_x|D-j 

X— d 

.-.  n — x 

1 

bArbAra 

darll 

3 

J-x,D-j 

n-d 

.-.  n— X 

m 

bArbAra 
or,  A,  A,  i 

davAptl  \ 
disAmIs  j 

4 

J-x    D-j 

D-n 

.-.  n— X 

(2)  (1) 
brAmAntip 

(3)  (4)      " 
dimArls 

bArbAra 

datlsl 

5 

j 

J-x 

D-j 

d— n 

.-.  n— X 

dArIi 

disAmIs 

1 

•^1 

J-x 

f^-j 

D-n 

.-.  n  —  X 

dImAris 

dimArls 

dArApti 

disAmls 

1 

D-x 

D-j 

J-n 

.-.  n  — X 

dArApti 

dimArls 

dAtlsi 

disAmla 

8 

D-x 

a-j 

J-n 

.-.  n  —  X 

dIsAmis 

dimArls 

dIsAmis 

disAmIs 

9 

d-x 

D-j 

J-n 

.-.  n  — X 

dAtlsi 

dimArls 

brAmAntip 

disAmls 

10 

X-d 

D-j 

J-n 

.*.  n — X 

I  bArbAra 
1  or,  A,A,i 

bramAntIp 
dimArls 

dim  Aril 

disAmIs 

11 

x-d 

D-j 

J-n 

.'.  n  —  X 

dArR 

dimArls 

cElArenf. 

celArEnt 

12 

J-X 

D-j 

X-d 

.-.N-X 

cAmEnes 

cesArE 

cEsAre 

cdArEnt 

13 

X-J 

D-j 

X-d 

.-.X-X 

cAmEstres 

cesArE 

14 

x-d 

D-j 

N-J 

.-.X-X 

bArbAra 

camEstrEs 

15 

x-d 

D-j 

J-X 

.-.X-X 

bArbAra 

camEnEs 

cAmEnes 

celArEnt 

16 

X— d   B--J 

N-j 

/.X-X 

cElArmt 

cesArE 

cAmEstrea 

celArEnt 

17 

X-j    &-J 

X^-d 

.-.X-X 

cEsAre 

cesArE 

SORITES. 


49 


of  Simple  Categorical  Sorites. 


Series  of  Stixogisms  j 
IS  WHICH  THE  Middle  ' 

i 

b 

O 

1 

1 

..^                    ^                                      Series  of   Stllogisms 

PROGRESSn-E   COXFIGUBATION.                        ^^,  ^^.^,^^  ^^^  ^^^^^ 

Premise 

Is     Mesor,  ' 

Magnts 
Major    of 

Premise     is     M  ajob, 

AND        THE 

Premise 
the  first 

Magnus 
Premise. 

Middle  '■ 
Premise, 

Maximus 
Premise, 

Ultima. 

AND     THE      Magnus 
Premise    Minor    op 

TUB  FIRST. 

N-d 

1^-J 

J-X 

.-.X-x 

(2)    (1) 
bArbAra 

(3)       (4) 
bArbarA 

hrAmAntip 

dAtisI 

2 

^X-d 

T^-J 

J-X 

.    ,,       V    ^bArbAra 
■'  ^^  —  ^      ,  or.  A,  A,i 

A,  a.  I 
dAril 

film  Arts 

dAmi 

3 

n-d 

I^-J 

J-X 

.'.  n  —  X  ,dArll 

dAril 

dArApti 

dAtUI 

1 
4 

1 

Di^n 

I>  — J 

J-X 

.'.  11  —  X  :  dArApfi 

\ 

dAril 

dhAmis 

dAtisI 

5 

d-n 

1^-J 

J-X 

.-.  n  — X    dAtisi 

dAril 

dAtM 

dAthI 

6 

D-n 

<^-J 

J-X 

•••  "*-^ 

dIsAmis 

dAnI 

hArbAra 
or,  A,  A,  i 

dAraptl) 
dAtlsI    S 

7 

J-n 

r>-j 

P-X 

.-.  n  — X 

brAmAntip 

dAril 

dArIi 

dAthI 

8 

J-n 

d— j 

D-X 

.*.  n — X  , 

dlniAris 

dAril 

hArbAra 

dlsa?nls 

9 

J-n 

i>-j 

d-X 

.*.  n — X 

bArbAra 

brAmantIp 

10 

J-n 

T>— j 

X  — d 

.-.  n  — X 

bArbAra 

dimarls 

11 

J-n 

T^-J 

x-d 

...„-.; 

!l2 

1 

X-d 

l>-j 

J-X 

.-.X-X 

bArbAra 

cElarErU 

1 
|13 

X-d 

D-J 

X-J 

.-.X-X 

bArbAra 

cEgarE 

cEsAre 

cAmenEg 

1 

|14 

^-J 

D-J 

X-d 

.•.5?-X 

cAmEstres 

cAmestrEs 

cElArent 

cAmmEs 

|l5 

J-K 

l>-j 

X-d 

.-.N-X 

cAmEnes 

cAmestrE8 

cAmEstres 

cAmenEa 

16 

^^-j 

&-J 

X-d 

/.x-x 

cEsAre 

cAmestrEs 

cAmEnes 

cAmenEs 

nix-d  &-J 

1 

.-.x-x 

cElArent 

i 

cAmestrEs 

50 


LOGIC   AS   A   PUKE   SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms 


Sbkies  op  Syllogisms 
in  which  the  middle 
Premise     is    Minor, 

AND      THE       MaXIMUS 

Pr/^mise    Major    op 
tus  first. 


cMArent 


e.  A,  O. 


cElArent  ferlO 

cElArent  ifelAptOn  I 

or,  E,  A,  o  ookArdO  j" 

cElArent  ferlsO 

fErlo  bokArdO 


fElApton 
fErIm 


bokArdO 


bokArdO 


hOkArdo      bokArdO 
fEsApo        bokArdO 


p 
§ 


Regressive  Configukation. 


frEsIson 

cEsAre 

cEsAre 


cEsAre 
or,  E,  A.o 


cBsAre 
fEstIno 


bokArdO 


i.  A,  0. 

1st  fig. 


ferlO 

fdAptOn  I 
bokArdO  ) 

ferlsO 

bokArdO 


18 
19 
20 

211 

I 
22: 

23 

24 
25 

m 

27 

28: 

29 

30 

31 
32 


Maximus 
Premise. 


J-X 

J-X 

^-X 
J-X 

&-X 

B^-X 
d-^X 

X-D 

X-D 

X-J 

X-J 

X-J 

X-J 
X-J 


Middle      Magnus 
Premise.   Premise. 


D-j  j  X-(l 

I 
D- 

D- 

D- 

d- 


D- 

d- 
D- 

d- 

D- 

D- 
D- 

D- 

d- 


11  — d 

D-11 

d-n 
D-n 

J-n 

J-n 
J-n 

J-n 

J-n 

X-d 

n-d 


Ultima. 


Series  op  Syllogisms 

IN  WHICH  THE  MlDDLE 

Premise  is  Major, 
AND  the  Maxim  ls 
Premise  Minor  of 
the  first. 


.-.  n-^X 


(2)(i)         j        (3)(4) 

I 


.*.  n— v^X  cAniEnes'  festlnO 
.'.  n—^X.  c^»iBnes  fesApO 
\  jl—^^\\cAmEn€8      fresIsOn 


:.  n-^X 

.-.  n-^X 

/.  n-^X 

.-.  n-^X 

.-.  n-^X 

/.  n-s^X 

/.  n^X 

cAmEstres 

e,A,  O. 


fis. 


n— v^X   cAmEstres  festInO 

D  — 11  I.*.  l\—^\'\cA7nEstres  fesAvO 

d  —  n  I .*.  n-wX  I  \cAmEstres  fresis fm 
D-n  /.  n-^Xii 


SORITES. 


61 


of  Simple  Categorical  Sorites.    (Continued.) 


Series  of  Syllogisms 

EN'  WHICH  THE  MrDDLE 

Precise    is     Mdjor, 

i 

5 

u, 

o 

1 

18 

Progressive  Contiguration. 

'!  Series  of  Syllogisms 

j     IN  which  the  Middle 

Premise    is     Major, 

AND       THE        MagNITS 

Premise    Major    or 

THE   FIRST. 

Magnus 
Premise. 

Middle 
Premise. 

Maximus 
Premise. 

Ultima. 

^       AXD       the        MagIOTS 

j     Premise     Minor    of 
I     THE  first. 

1 

brAf/iAntip 

fErisO 

N^-d 

D-j 

J-X 

.-.  n-^X 

1      (2)  (1) 
J  bArbAra 
ioT,A,A,t 

(3)       (4) 
E,a,  0 
fEriO 

(ITrnAr'is 

fEHsO 

19 

n-d 

n-j 

J-X 

.-.  n-^X 

dArR 

fEriO 

(lArApfi 

fErisO 

20 

D-n 

D-j 

J-X 

.-.  n-^X 

dAvApti 

fEriO 

d  Is  Amis        fErisO 

21 

d-n 

ii-j 

J-X 

.-.  n-^X 

dAtM 

fEHO 

(lAtlsi           fErUO 

22 

D-n 

'i-j 

J-X 

.-.  n-^X 

dig  Amis 

fEnO 

bArbAra      /ElaptOn  1 
or,  A,  A,  i     fErisO      1 

23 

J_n    D  — j 

D-X 

.'.  n-wXi  brAmAntip 

fEHO 

dArIi            fErisO 

24 

J-n 

d-j 

D-X 

.:  n-^X 

dim  Alia 

fEHO 

bArbAra      bOkardO 

25 

J-n 

D-j 

d^X 

.-.  n^X 

1 

bArbAra      fE^apO     \ 
or,  A,  A,  i    frEsisOn  f 

26 

J-n 

l>-j 

X-D 

.*.  n-^X 

brAmAntip 

fEstinO 

dArR           frEsisOn, 

27 

J-n 

d-j 

3t-D 

.-.  n-^X 

dImAris 

fEstinO 

brAmAntip  frEsisOn 

28 

N— d 

D-j 

X-J 

/.  n-^X 

1 

1 

J  bArbAra 
\oT.A.A.i 

E,a,  0 
fEstinO 

dim  Arts 

frEsisOn 

29 

n-d 

D-j 

X-J  i 

.-.  n-^X 

dArR 

fEstinO 

'!  U'Apfi        irEsisOn     \ 

\ 

30 

D-n 

D-j 

X-J 

.-.  n-^X 

dArApti 

fEstinO 

I 
'UsAmis       frEHsOn 

31 

d-n 

D-j 

X-J 

/.  n-^X 

dAtIsi         \ 

fEstinO 

1 
dAilH          frEsisOn 

1                     i 

32 

D-n 

d-j 

5-.JI 

.-.  n-^X 

dIsAmis 

1 

fEstinO 

52 


LOGIC   AS  A  PURE   SCIENCE. 


Synopsis  of  all  Possible  Valid  Forms 


Series  of 

IN  WHICH  1 

Premise 

Syllogisms 
rHE  Middle 
IS     Minor, 
Maximus 
Major    of 

Ph  ! 

O   1 
m 

33 

Kbgressive  Configuration. 

Series  of 

IN  WHICH 

Premise 

AND      THI 

Premise 

THE   FIRS 

Syllogisms 

THE  Middle 

IS    Major, 

AND      THE 

Premise 

THE  FIRST 

Maximus 
Premise. 

Middle 
Premise. 

Magnus 
Premise. 

Ultima. 

:     Maximus 
Minor    of 
r. 

^AmEnes 

j  e.  A,  0.  \ 
1     1st  fig.  f 

X-d 

&-J 

^-} 

.-.  n-^X 

'  (2)(1) 
cElArent 

(3)  (4) 
i  e.  A,  0. 
1         adfio; 

cAmEnes 

ferlO 

34 

X-d 

B--J 

n-j 

.-.  n-^X 

cElArent 

festInO 

cAmEnes 
or,  A,  E,o 

felAptOn  ) 

35 

X-d 

^-J 

J-n 

/.  n-^X 

cElArent 

fesApO 

xAmEnes 

ferlsO 

36; 

X-d 

B-J 

j-^i 

.-.  n-^X 

CElArent 

fresIsOn 

37 

X-d 

l)-j 

5^-J 

/.  n^X 

bArbAra 

j  a,  E,  0. 
1        2d  fig. 

38 

X-d 

D-J 

n-wJ 

/.  n-^X 

bArbAra 

bar  Ok  0 

39 

X-d 

D-J 

J-X 

/.  n^X 

bArbAra 

(  a,  E,  0. 

»       4th  fig. 

cAmEstres 

1    let  fig.  f 

40 

X-j 

^-J 

X-d 

/.  n-^X 

cEsAre 

J  e,  A.  0. 
1         2d  fig. 

cAmEstres 

ferlO 

41 

X-j 

&-J 

n~d 

.-.  n-^X 

cEsAre 

feotInO 

cAmEstres 
or,  A^E^o 

felAptOn  1 
bokArdO   f 

42' 

X-j 

^-J 

D-n 

.-.  n-^X 

cEsAre 

fesApO 

cAmEstres 

ferlsO 

43 

x-j 

i>-J 

d-n 

cEsAre 

free  Is  On 

hArOko 

bokAr.iO 

44 

x-j 

I 

d-^J 

D-n 

.-.  n-^X 

1 

SORITES. 


53 


of  Simple  Categorical  Sorites.    (Concluded.) 


Series  of  Stixogisms 

IN  WHICH  THE  MIDDLE 

i: 

Progressive  Configuration. 

Series  op  Syllogisms 

IN  WHICH  THE  MIDDLE 

Premise     is    Mijjob, 

o 

Premise 

AND        TH 

IS    Major, 

AND      THE     Magnus 

E      Magnus 

Premise    Major    of 

Tj    *■ 

Magnus      Middle 

lilaximus 

Ultima. 

Premise 

Minor    op 

THE  FIRST. 

O 
^  ■ 

33 

Premise.   Premise. 

Premise. 

THE  first. 

cAmEstres 

j  A\  e,  0.  ] 
1   4th  fig.  f^ 

i 

N-j    B--J 

X-d 

j 

J  cEsAre 
1  or,  E,  A.  0 

^3)    (4) 
A,e,  0 
bArokO 

1 

U 

n_j  1  &- J 

X-d 

.-.  n-^x; 

fEstIno 

bArokO 

1 
1 

35 

J_n    &-J 

x-d 

.-.  11-^x' 

fEsApo 

bArokO 

30 

j_n    B--J 

X-d 

.-.  n_Xi 

frEsIson 

bArokO 

cEsAre 

S  A,  e,  0. 
\       4th  fig. 

37 

5^_J   D-j 

x-d 

.-.  ii-^X 

J  cAmEstres  A.  e.  0 
j  or,  A.E,o    bAjvkO 

38 

n-^JjD-j 

x-d 

.',  n^X 

hArOko 

bArokO 

cMArent 

\  A,  e,  0. 
\        4th  fig.! 

39 

J-N 

D-J 

x-d 

.-.  n-s^X 

j  cAmE'ies 
\  or,  A,E,o 

A,  e.  0 
bArokO 

cAmEnes 

)  A,  e,  0. 
\       4th  fig. 

40 

N-d 

&-J 

x-j 

.-.  n-^X 

J  cElArent 
\  or,  E,A.o 

A.e.  0 
bArokO 

41 

n-(l 

B--J 

x-j 

.-.  n-wX 

fErlo 

bAwkO 

1 

42 

D-n 

&-J 

x-j 

.-.  n-^X 

fElApton 

bArokO 

43 

d-ii 

]&-J 

x-j 

.-.  n-^X 

fErleo 

bArokO 

j 

44 

D-n 

d-^J 

x-j 

••— " 

bOkArdo 

bArokO 

64  LOGIC   AS  A  PURE  SCIENCE. 

§  11.  The  number  of  forms  of  valid  Sorites,  shown  in 
the  foregoing  synopsis,  is  eighty-eight,  forty-four  on  each 
side  ;  but  a  comparison  of  them,  line  by  line,  read  across 
both  iDages  of  the  synopsis,  will  show  that,  considered 
with  respect  to  the  propositions  of  which  they  are  com- 
posed, without  regard  to  the  order  of  their  statement, 
there  are  but  forty-four  ;  the  first  and  third  propositions 
in  the  regressive  configuration  changing  places  in  each 
case,  and  becoming  respectively  third  and  first  in  the 
progressive  throughout  the  whole  series,  the  middle 
premise  and  ultima  being  the  same  in  each  case  on  both 
sides  throughout.  They  are  numbered  from  one  to  forty- 
four,  on  each  side,  to  correspond. 

To  one  or  another  of  these  forms,  EYElfY  valid  argu- 
ment (expressed  categorically)  involving  four  terms,  or, 
as  will  be  hereinafter  shown,  involving  any  greater  num- 
ber of  terms,  MUST  BE  conformed. 

§  12.  The  moods,  as  determined  by  the  quantity  and 
quality  of  the  propositions  (indicated  by  their  symbols), 
are  twenty  in  number,  of  which  fourteen  occur  in  both 
configurations,  three  in  the  regressive  only,  and  three  in 
the  progressive  only. 

The  following  table  shows  them,  arranged  in  the  order 
A,  I,  E,  O  of  the  symbols  of  the  ultima,  with  their  num- 
bers in  each  configuration,  as  in  the  synopsis,  repeated 
where  they  are  both  minor  and  major.  The  symbols  are 
in  capitals  in  the  synopsis,  the  first  two  in  the  columns 
of  Syllogisms,  on  the  right-hand  side  of  each  page 
{majors)  being  transposed,  as  previously  stated,  and  as 
shown  by  the  figures  over  those  columns. 


SO«ITES 


55 


Moods  of  Sorites. 


1                       i 

Nos.  m  Regressive  Conpigubation,     Nos.  ts  Progbessite  Configuration. 

Symbol?. 

ij 

Minors. 

Majors.           J           Minors.                       Majors. 

A,  A,  A,  A. 

1. 

1. 

A,  A,  A,  I. 

2,  4,  7,  10. 

4,  7,  10. 

2,  4,  7,  10. 

2,  4,  7. 

A,  A,  I,  I. 

3,  5. 

9,11. 

1,  A,  A,  I. 

9.  11. 

9,11. 

3,5. 

3,5. 

A,  I,  A,  I. 

6,8. 

6,8. 

6,  8. 

6,8. 

E,A,A,E. 

12,  13. 

12,  13. 

14,  15. 

14,  15. 

A,  A,  E,  E. 

14,  15. 

12,  13.  ^ 

16,  17. 

A,  E,  A,  E. 

16,  17. 

16,  17. 

16,  17. 

E,  A,  A,  0. 

J 18,  20,  23, 1 
|26,  28,  30.  j" 

18,  20,  28,  30. 

37,  39. 

37,  39. 

A,  A,  E,  0. 

37,  39. 

jl8,  20,  23,  ( 
126,  28,  30.  i 

j  18,  20,  23, ) 
126,  28,  30.  (■ 

E,  A,  I,  0. 

19,  21,  29,  31. 

19,  21,  29,  31. 

I,  A,  E,  0. 

19,  21,  29,  31. 

19,  21,  29,  31. 

E,  L  A,  0. 

22,  24,  27,  32. 

A,  I,  E,  0. 

22,  24,  27,  32. 

22,  24,  27,  32. 

0,  A,  A,  0. 

25.    • 

38. 

A,  A,  0,  0. 

38. 

25. 

A,  E,  A,  0. 

33,  35,  40,  42. 

33,  35,  40,  42. 1 

33,  40. 

33,  35,  40,  42. 

A,  E,  I,  0. 

34,  36,  41,  43. 

34,  36,  41,  43. 

I.  E,  A,  0. 

34,  36,  41,  43. 

A.  0,  A,  0. 

44. 

44. 

66  LOGIC   AS   A   PURE   SCIENCE. 

It  is  manifest  that  it  would  be  a  very  difficult  thing 
to  classify  Sorites  in  figures,  according  to  the  positions 
of  the  terms,  and  to  devise  names  for  the  moods,  anal- 
ogous to  those  of  simple  Syllogisms ;  and,  if  it  should 
be  accomplished,  the  figures  and  names  of  the  moods 
would  be  extremely  burdensome  to  the  memory.  The 
different  forms  can  be  much  more  readily  referred  to 
by  their  numbers  and  the  names  of  the  configurations, 
as  adopted,  than  by  their  symbols,  or  any  names  that 
could  be  devised  for  them.  They  will  be  hereinafter  so 
referred  to. 

By  counting  the  series  of  Syllogisms  on  the  left 
{minors)  and  right  {majors)  of  the  synopsis  in  each  con- 
figuration, there  will  be  found  to  be  : 

J^  Regressives.  Progressives. 

Minors.  Majors,  Minors.  Majors. 

39.  '62.  33.  40. 

corresponding  to  the  numbers  shown  in  the  table  on 
page  46. 

§  13.  Sorites,  in  the  regressive  configuration,  may  be 
expanded  into  series  of  Syllogisms  in  all  combinations  of 
figures,  except  those  of  the  third  and  first,  and  third  and 
second  ;  and  those  in  the  progressive  configuration,  in  all 
combinations,  except  those  of  the  second  and  first,  and 
second  and  third. 

Such  of  them  as  can  be  expanded  wholly  in  the  first 
figure,  are  the  only  perfect  forms.  The  series  of  Syllogisms, 
in  which  they  can  be  so  expanded,  occur  in  the  synopsis 
only  on  the  left  side  of  the  regressives  {minors),  and  on 
the  right  side  of  the  progressives  {majors) ;  and  the  first 


SOEITESo  57 

figure  occurs  as  the  figure  of  the  second  Syllogism  only 
on  the  same  sides.  Moods  Nos.  10,  11,  15,  26,  27,  35, 
36,  and  39  cannot  be  expanded  directly  (that  is,  without 
conversion)  except  by  the  aid  of  the  fourth  figure ;  a 
fact  which  may  tend  in  some  measure  to  relieve  that 
figure  from  the  odium  which  has  been  cast  upon  it. 

§  14.  Tliere  is  a  very  remarkable  and  wonderful  anal- 
og}^ between  the  forms  of  reasoning  and  the  two  simplest 
forms  of  geometrical  figures,  plane  and  solid  (with  plane 
surfaces) ;  an  analogy  which  is  evidently  something  more 
than  merel}'  fanciful. 

The  Syllogism  of  logic  and  the  triangle  of  geometry, 
and  the  Sorites  and  tetrahedron  are,  respectively,  similar. 

The  triangle  consists  of  three  points,  equivalent  to 
the  three  terms  of  the  Syllogism,  connected  by  three 
lines,  which  answer  to  the  copulas  of  the  propositions. 
l\o  plane  surface  can  be  represented  by  less  points  and 
lines,  no  argument  by  less  terms  and  propositions.  By 
means  of  the  former,  with  the  aid  of  the  latter,  all  phys- 
ical relations  in  space  are  determined,  not  only  on  the  sur- 
face of  the  earth  and  within  it,  irom  those  of  the  smallest 
subdivision  to  those  of  continents  and  oceans,  but  also  in 
the  heavens  to  the  remotest  star-depths,  so  far  as  the 
stars  can  be  brought  under  observation ;  by  the  latter 
all  relations  are  determined,  not  only  of  physical  things, 
but  also  of  the  metaphysical  and  immaterial.  But  the 
analogy  does  not  end  here.  In  its  very  practical  con- 
struction the  triangle  produces  the  equivalent  of  a  per- 
fect Syllogism  in  Barbara.  If  we  are  at  any  point,  N, 
on  the  surface  of  the  earth,   from  which  we  can  see 


68  LOGIC   AS   A  PURE   SCIEISCE. 

another  point,  J  (either  on  the  earth  or  in  the  heavens), 
which  is  inaccessible,  and  the  distance  to  which  we  can- 
not therefore  directly  measure,  we  may  select  another 
point,  D  (either  on  the  earth  or  its  orbit),  which  is  acces- 
sible, and  from  which  the  point,  J,  may  also  be  seen ; 
and  first,  carefully  observing  the  directions  from  N  to  J, 
and  from  N  to  D,  and  thus  determining  the  angle,  we  may 
then  proceed  to  measure  the  distance  between  N  and  D 
in  a  straight  line.  The  line  thus  laid  down  is  equivalent 
to  the  first  proposition,  IS"  —  d,  with  which  we  set  out  in 
§  2  of  this  chapter.  Arrived  at  D,  we  may  then  observe 
the  direction  from  D  to  J,  and  determine  the  angle,  and 
then,  by  means  of  the  elements  thus  obtained,  we  may 
determine  the  distance  in  a  straight  line  from  D  to  J,  and 
from  N  to  J.  The  lines  thus  drawn,  or  supj)osed  to  be 
drawn,  are  the  equivalents  of  the  second  proposition, 
D  —  j,  with  which  we  set  out,  and  of  the  conclusion  to  be 
deduced  from  it  and  the  first  proposition,  IS"  —  d,  when 
put  forth  as  premises  of  a  Syllogism,  namely,  'N  —  j. 

The  tetrahedron  is  the  simplest  form  in  which  any 
solid  with  plane  surfaces  can  be  included,  and  is  the 
analogue  of  the  Sorites.  Its  four  points  answer  to  the 
four  terms,  its  four  planes  (each  in  the  form  of  a  triangle) 
bounded  by  six  lines  (each  being  a  boundary  of  two 
planes)  to  the  four  Syllogisms  of  the  two  principal 
series ;  each  series  with  its  six  propositions.  Each  plane 
connects  three  points,  each  Sjilogism  three  terms.  Each 
of  the  four  points  is  excluded  from  one  of  the  planes, 
each  of  the  four  terms  from  one  of  the  Syllogisms. 
To  illustrate  by  means  of  geometrical  figures  : 
If  we  take  a  piece  of  card-board  and,  having  cut  it  in 


SORITES.  69 

the  form  of  an  equilateral  triangle,  inscribe  therein 
another  equilateral  triangle,  the  lines  of  which  terminate 
in  the  middle  of  the  lines  of  the  exterior  one,  and  mark 
all  the  angles  with  letters,  as  follows  : 


we  may  then  fold  the  card-board  backward  on  the  lines 
of  the  inscribed  triangle  so  as  to  bring  together  the  three 
points,  X,  X,  X,  and  then  fastening  together  the  edges 
of  the  card -board  so  brought  together,  we  shall  have  a 
regular  tetrahedron,  the  very  embodiment  of  a  simple 
Sorites.  Looked  at  from  our  present  stand-point,  we 
shall  see  only  the  inscribed  triangle  No.  3,  and  having  its 
angles  marked  with  the  letters  N,  D,  and  J.  The  other 
triangles  and  the  point  X  will  not  be  seen.  Turning  the 
figure  about,  so  as  to  bring  its  planes  before  us  in  the 
order  in  which  they  are  numbered,  and  considering  them 
in  two  series  of  two  each,  we  shall  find  them  as  follows, 
beginning  at  the  right  hand  with  the  first  series,  and 
reading  backward,  but  from  left  to  right,  in  the  second. 


60 


LOGIC   AS  A   PURE   SCIENCE. 

First  series. 


Second  series. 


Observing  that  the  letters  at  the  apices  of  the  tri- 
angles are  the  middle  terms  of  the  Syllogisms  of  the  two 
principal  series  hereinbefore  shown,  and  considering  the 
lines  of  the  triangles  as  copulas  connecting  the  terms  of 
propositions,  and  the  lines  at  the  bases  as  indicating  con- 
clusions, and  beginning  with  the  first  series  of  triangles 
at  the  right  hand  and  regressing^  we  can  read  as  follows : 

Because  D  is  J  and  J  is  X,  therefore  D  is  X  ;  and  because  N  is  D  and 
D  is  X,  therefore  N  is  X, 

and  then  going  to  the  second  series,  and  beginning  at  the 
left  hand  and  progressing,  we  can  further  read  : 

Because  N  is  D  and  D  is  J,  therefore  N  is  J ;  and  because  N  is  J  and 
J  is  X,  therefore  N  is  X. 

The  correspondence  between  the  triangles  and  the 
Syllogisms  is  exact  throughout,  except  that  the  premises 


SORITES. 


61 


in  the  latter  are  transposed,  but  the  order  of  statement 
of  the  premises  is  a  matter  of  no  consequence,  the  terms 
determining  their  character. 

The  middle  terms  D  and  J  may,  of  course,  be  tmns- 
posed  in  our  original  illustration,  and  in  such  case  the 
numbers  2  and  4  would  also  have  to  be  transposed,  and 
the  i:)ositions  of  all  the  letters  and  the  numbers  in  tri- 
angles 1  and  3,  relatively  to  the  whole  figure,  would 
also  require  to  be  changed.  The  first  series  of  triangles 
would  then  read  forward  and  the  second  backward,  but 
the  series  of  Syllogisms  would  remain  the  same,  the 
first  regressive,  and  the  second  progressive. 

The  four  triangles  may  also  be  exhibited  in  the 
following  form : 


and  may  be  folded  on  the  interior  lines  with  like  result 
as  before. 

But  the  Sorites  is  superior  to  its  analogue,  the  tetra- 
hedron, in  this,  that  its  ultimate  conclusion  is  reached  by 
either  process,  regressive  or  progressive,  but  both  are 
required  to  complete  the  tetrahedron:  This  will  be 
apparent  by  the  consideration  of  the  two  following 
forms. 


62 


LOGIC   AS  A  PUKE  SCIENCE. 


/  \ 
/       N 


If,  in  the  first,  beginning  with  'N,  we  successively 
reach  by  investigation  the  points  D,  J,  and  X,  and  then 
commence  to  reason  with  the  propositions  which  we 
enounce  as  the  results  of  our  investigation,  we  may  by 
two  Syllogisms,  of  which  the  two  completed  triangles 
1  and  2  are  analogues,  arrive  at  the  ultimate  conclusion. 
But  if,  in  the  second,  by  the  same  process  of  investiga- 
tion we  reach  only  to  the  point  J,  and  then  commence 


SORITES.  63 

to  reason,  we  frame  our  first  Syllogism,  of  which  the 
triangle  3  is  the  analogue,  resulting  in  the  conclusion 
that  N  —  J.  We  are  thereupon,  if  we  would  advance^ 
further,  obliged  to  resume  investigation,  and  through  it 
reach  out  to  X,  and  are  thence  enabled  to  frame  the 
second  Syllogism,  of  which  the  triangle  4  is  the  analogue,  •■ 
arriving  at  the  same  ultimate  conclusion.  But  in  either 
case  the  tetrahedron  is  incomplete,  and  can  only  be  com- 
pleted by  the  union  of  the  two.  Each  figure  is  the  com- 
plement of  the  other,  required  to  make  the  perfect  figure, 
shown  in  our  first  illustration. 

But  again,  the  two  different  processes,  regressive  and 
progressive,  in  respect  to  argumentation  by  Syllogisms, 
are  analogous  to  the  two  possible  combinations  of  the 
two  processes  by  which  we  may  determine  the  length  of 
the  concluding  line  with  which  we  enclose  a  triangle. 

Leaving  N,  and  going  to  D,  we  observe  the  direction 
in  which  we  are  traveling,  and  measure  the  distance 
tmveled.  Then  observing  the  direction  from  D  to  J, 
and  thus  determining  the  angle,  we  go  on  from  D  to  J, 
measuring  the  distance.  If  we  then  stop,  we  may,  by 
the  three  elements  thus  obtained,  viz.,  the  two  lines  and 
the  included  angle,  determine  the  distance  and  direction 
from  N  to  J ;  then,  having  this  distance  and  direction, 
and  observing  the  direction  of  X  from  J,  we  go  back  to 
N,  and  observe  its  direction  from  X,  and  determine  the 
angles,  and  then  with  the  three  elements  thus  secondly 
obtained,  viz.,  the  two  angles  and  the  included  line  from 
N  to  J,  we  may  determine  the  distance  from  N  to  X. 
This  is  analogous  to  the  progressive  process. 

But  if,   after  reaching  J,   without   stopping  to  de- 


64  LOGIC    AS  A   PURE   SCIENCE. 

termine  its  distance  from  'N,  we  observe  the  direction 
therefrom  to  X,  as  in  triangle  1,  and  going  back  to  D, 
observe  also  its  direction  from  X,  and  determine  both 
angles,  then  with  the  three  elements  thus  obtained  (being 
like  to  those  of  the  second  three  in  the  preceding 
process),  we  may  determine  the  distance  from  B  to  X, 
and  then,  having  the  distances  and  directions  from  D  to 
IS",  and  from  D  to  X  (the  figure  being  now  considered 
as  folded),  and  determining  the  included  angle,  we  may 
by  such  elements  (being  like  to  those  of  the  first  three 
in  the  preceding  process)  determine  the  distance  from 
N  to  X.     This  is  analogous  to  the  regressive  process. 

Surely,  in  all  this  wonderful  accord  there  must  be 
something  more  than  mere  coincidence.  ''The  invisible 
things  of  God  are  clearly  seen,  being  perceived  through 
the  things  that  are  made." 

§  15.  But  the  subject  concerning  which  we  set  out  to 
make  investigation  may  be  the  summum  genus  instead 
of  the  infima  species  or  individual,  as  hitherto,  and  in 
such  case  we  shall  find  that  the  processes  of  both  inves- 
tigation and  reasoning  will  be  in  the  exactly  opposite 
direction,  and  that  the  maximus  term,  instead  of  the 
magnus,  as  hitherto,  will  be  the  subject  of  the  ultima, 
and  the  magnus  term  instead  of  the  maximus  will  become 
the  predicate. 

Strictly  speaking,  the  word  "predicate"  is  not  prop- 
erly applicable  to  the  last,  but  rather  to  the  first  term  of 
propositions  as  they  will  be  exhibited  in  this  section, 
inasmuch  as  the  species  cannot  be  predicated  of  the 
genus,  but  the  genus  of  the   species.     To  change  the 


SOEITES. 


Qd 


names  of  the  terms  as  they  stand  related  to  the  proposi- 
tions would,  however,  be  confusing,  and  they  \vill,  there- 
fore, be  retained  in  their  grammatical  rather  than  in 
their  strict,  logical  signification. 

But  we  shall  find  it  necessary  to  change  the  signifi- 
cation of  the  copula.  As  hitherto  employed,  such  sig- 
nification has  been  '*is"  or  '"is  not"  in  the  sense  of  ""is 
(or  is  not)  coinprehended  in,^^  but  as  employed  in  this 
section  ohly,  the  copula  must  be  understood  to  signify 
''comprehends^'^  or  ''does  not  comprehend.'^  The  rea- 
son for  this  change,  if  not  immediately  obvious,  will 
become  clear  as  we  jjrogress.  It  will,  however,  be  here- 
after seen  that  in  some  cases  the  two  significations  are 
interchangeable,  and  either  may  be  understood. 

I  shall  have  immediate  recourse  to  illustration  by 
means  of  geometrical  figures,  as  thereby  such  illustmtion 
can  be  made  much  clearer,  being  exhibited  to  the  eye  as 
well  as  to  the  understanding ;  and  I  now  give  the  fol- 
lowing fiarure. 


66  LOGIC   AS   A   PURE  SCIENCE. 

which  you  will  observe  is  like  to  our  original  card- 
board figure  on  page  59,  with  triangles  1  and  3  remain- 
ing in  the  same  position  as  therein,  but  with  triangles 
2  and  4. turned  upward,  each  in  a  semicircle,  on  the 
points  D  and  J  as  centres  respectively. 

The  jioints  XXX  are  now  brought  together  in  the 
figure,  and  N  JS"  N  separated  and  become  exterior.  The 
X)oints  D  D  D  and  J  J  J  retain  their  intermediate  posi- 
tions. 

If  now  we  begin  to  make  investigation  concerning  X 
as  the  subject,  we  shall  find  ourselves  proceeding  in  a 
descending  instead  of  ascending  direction,  as  before  ;  and 
we  shall  also  find  that  the  notions  which  we  discover  as 
predicable  (in  the  sense  of  the  copula,  as  above  changed), 
of  our  successive  subjects,  instead  of  being  higher  genera 
and  comprehending  the  subjects,  are  lower  species,  and 
are  wholly  comprehended  in  the  subjects  respectively. 
The  propositions  in  which  we  lay  down  our  judgments 
will  therefore  necessarily  be  required  to  signify  this  dif- 
ference, which  may  be  done  by  j)utting  the  predicates  in 
capitals  instead  of  small  letters,  as  before,  and  will  be  as 
follows : 

X  ■: —  J  {meaning  All  X  comprehends  all  J)  ; 
J  —  D  {meaning  All  J  comprehends  all  D) ; 
D  —  N   {meaning  All  D  comprehends  all  N). 

The  subject  of  each  of  the  foregoing  propositions  is 
distributed.  But  it  might  have  been  undistributed  in 
so  far  as  relates  to  the  manner  of  its  representation,  and 
the  proposition  still  retain  its  character  as  universal. 

To  illustrate,  I  now  reproduce  the  first  combination  of 
circles  shown  in  the  former  part  of  this  treatise  when 


SORITES. 


67 


treating  of  simple  Syllogisms,  adding  another  circle  to 
make  it  applicable  to  a  Sorites,  the  letters  being  put  on 
the  lines  of  the  circles,  and  to  be  considered  as  indicating 
the  whole  areas  included  in  the  circles  respectively. 


It  will  now  be  manifest,  from  mere  inspection  of  the 
figure,  that  what  we  have  predicated  of  X  (viz.,  J)  might 
also  have  been  predicated  of  x,  and  in  fact  with  more 
correctness,  for  J  is  comprehended  wholly  and  only  in 
tliat  part  of  X  which  lies  within  the  circle  marked  on 
one  side  J  and  on  the  other  x.  In  like  manner,  what  we 
have  predicated  of  J  (viz.,  D)  might  have  been  predi- 
cated of  j,  and  what  we  have  predicated  of  D  (viz.,  N) 
might  have  been  predicated  of  d. 

The  propositions  may  therefore  be  stated  as  follows  : 

X  or  X  —  J  ; 
J  or  j  —  D  ; 
D  or  d  —  K. 

In  either  alternative  the  propositions  must  be  re- 
garded as  universal.     I  shall  hereafter  make  use  only 


68  LOGIC    AS   A   PURE   SCIENCE. 

of  that  in  which  the  subjects  are  represented  by  small 
letters,  as  apparently,  but  not  in  fact,  undistributed.  In 
reading  the  propositions,  the  words  "All"  and  ''Some" 
must  be  expressed,  and  it  must  be  borne  in  mind  that' 
the  word  "Some"  applies  to  a  definite  part  of  the 
term,  and  when  in  the  process  of  the  reasoning  a  tenii 
with  that  word  prefixed  shall  be  repeated,  it  must  be 
read  or  understood  as  "The  same  some,"  or  "The 
same  definite  part  of." 

The  dictum  of  Aristotle,  as  applicable  to  the  above 
propositions,  will  now  have  to  be  changed  so  as  to  read 
as  follows. 

Whatever  definite  term  is  afiirmed  or  denied  as  com- 
prehending any  other  definite  term,  may  be  afiinned  or 
denied  as  comprehending  any  definite  term  compre- 
hended in  the  definite  term  so  comprehended,  and  in 
like  manner  of  any  definite  term  comprehended  in  the 
definite  term  so  secondly  comprehended,  and  so  on  ad 
infinitum. 

Applying  the  dictum  as  thus  changed  to  the  above 
propositions,  the  two  forms  of  the  full  Sorites  warranted 
thereby  will  be  as  follows  : 


In  the  regressive 
configuration. 

In 

the  progressive 
configuration. 

d  —    N, 

X-     J, 

j  -    D, 

J-D, 

X-    J; 

d  —  N; 

.-.  X  —    N. 

• 

•.  X  -  N, 

and  the  abridged  form  will  be 

•••  j  — 

D;    .-. 

X    - 

-  K. 

SOKITES.  69 

All  propositions  put  forth  in  the  above  form  in  the 
descending  processes  of  investigation  and  reasoning, 
may  be  converted  simply,  provided  the  original  signifi- 
cation of  the  copula  be  at  the  same  time  reinstated,  and 
by  simple  conversion  of  the  above,  we  shall  have  the  two 
forms  of  Sorites  as  we  have  hereinbefore  seen  them. 

But  not  only  have  the  terms  of  all  the  propositions  in 
the  two  forms  changed  places,  but  also  the  forms  them- 
selves, in  respect  to  the  configurations,  the  converse  of 
that  which  before  was  regressive  having  become  progress- 
ive and  of  that  which  was  progressive,  regressive.  By 
examining  our  original  card-board  figure  in  connection 
with  the  figures  on  page  62,  and  the  remarks  on  the  lat- 
ter, and  comparing  them  with  the  fii-st  figure  in  this  sec- 
tion, and  applying  such  remarks  to  the  configurations  as 
herein  gi'ven,  it  will  be  seen  that  such  change  is  proper, 
triangles  3  and  4  in  the  latter  figure  being  the  analogue 
of  the  Sorites  in  the  regressive  configuration,  and  1  and 
2  of  that  in  the  progressive. 

In  like  manner,  it  will  be  found  that  in  all  matters  of 
form  there  will  be  continued  inversions. 

The  Sorites  herein  given  may  be  expanded  into  series 
of  Syllogisms  as  follows  : 

In  the  regressive  process. 
j  -  D,  X  -   J, 

d-N;  j-N; 

.-.   j  —  N. --^^^^        .'.  X  —  N. 

In  the  progressive  process. 

X  —   J,  ^ X  —  D, 

j  -  D  ;  X^  d  -  N  ; 

.-.   X  —  P  -^  .'.  X  —  N. 


70 


LOGIC   AS   A  PURE  SCIENCE. 


All  the  propositions  in  the  foregoing  forms  are  uni- 
versal, but  they  may  all  be  particular  in  the  manner  of 
their  representation  (indicated  by  the  apparent  non-distri- 
bution of  the  predicate),  provided  the  definiteness  of  the 
terms  represented  be  kept  in  view.  Thus,  in  the  fol- 
lowing figure,  let  the  letters  on  the  lines  of  the  circles 
refer  to  the  whole  areas  of  the  circles  respectively  as 
before,  and  those  in  areas  only  to  the  areas  as  bounded 
by  lines  respectively,  but  considering  them  where  occur- 
ring more  than  once  as  to  be  taken  together : 


The  Sorites  exemplified  will  be  as  follows : 


In  the   regressive 

In 

the  2)?'ogressive 

configuration. 

configuration. 

d  —  11, 

X  —   J, 

J  -  d, 

J  -  d, 

X  —  j; 

d  -  n  ; 

.'.    X    11. 

.-.   X  —  11  ; 

and  may  be  expanded  into  series  of  Syllogisms,  as  fol 
lows: 


SORITES.  71 

In  the  regressive  process. 

j  —  <^,  X  —    j, 

a  —  n  ;  ^ j  —  n  ; 

.-.  j  —  11. .*.  X  —  n. 

In  the  progressive  process. 

X  —   j,  ^ X  —  d, 

j  —  d;  ^y^  d  —  n; 

.-.  X  —  d.  — ^'^'"^  .*.  X  —  11. 

Here  apparently  we  have  two  anomalies — viz.,  Syllo- 
gisms having  the  middle  terms  undistributed  in  both 
premises,  and  Syllogisms  in  which  conclusions  are  de- 
duced from  particular  premises.  But  they  are  such 
only  in  appeaiTince  ;  all  the  propositions  (the  definiteness 
of  the  terms  being  kept  in  mind)  being  in  fact  universal, 
and  the  middle  term  distributed  in  each  case  in  the 
major  premise. 

The  teiTiis  of  all  the  foregoing  propositions  may 
each  be  considered  as  comprising  all  the  areas  marked 
in  the  figure  ^\'ith  the  small  letters  representing  them 
respectively,  taken  together  respectively^  or  only  those 
areas  respectively,  in  which  the  letters  representing 
both  the  subject  and  predicate  appear,  taken  together. 
In  the  former  case,  the  subjects  will  each  be  greater 
than  their  predicates  respectively,  and  the  copula  must 
signify  ''^  comprehends,'"  in  the  latter,  the  terms  of 
each  proposition  will  be  co-extensive,  and  the  copula 
may  have  either  signification.  But  in  the  latter  case,  the 
major  middle  term  in  the  middle  premise  will  narrow  in 
signification  to  that  of  the  minor-middle  term,  and  the 
maximus  term  in  the  ultima  will  have  a  narrower  sig- 
nification than  as  employed  in  the  maximus  premise. 


72  LOGIC   AS   A   PURE   SCIENCE. 

The  middle  premise,  it  will  be  seen,  has  become  the 
major  premise,  and.  the  magnus  premise  the  minor  of  the 
first  Syllogism  of  the  series  in  the  regressive  process, 
and  the  middle  premise  has  become  the  minor,  and  the 
maxim  us  the  major  of  the  first  Syllogism  of  the  series  in 
the  progressive.  The  regressive  Sorites  in  the  descending- 
process  is  therefore,  in  the  forms  above  given,  which  you 
will  find  are  the  perfect  forms,  a  major  Sorites,  instead  of 
a  minor,  and  the  progressive  a  minor  Sorites  instead  of  a 
major  as  before.  It  will  be  also  seen  that  the  Enthymeme 
taken  from  the  second  Syllogism  in  the  regressive  series 
is  of  the  second  instead  of  the  first  order  as  before,  and 
vice  versa  in  the  progressive.  If  a  synopsis  should  be 
made,  this  would  necessitate  (to  make  it  conform  to  the 
former)  the  transfer  of  the  headings  of  the  columns  on 
each  side  of  each  page  of  the  former  from  each  page  to 
the  other,  and  their  transposition  after  being  so  trans- 
ferred. 

All  the  Syllogisms  are  in  the  fourth  figure,  which  in 
this  process  becomes  the  perfect  figure,  the  first  be- 
coming imperfect.  The  second  and  third  figures  will 
also  be  found  to  have  changed  places,  if  indeed  they 
and  the  first  can  have  any  place  at  all,  in  the  new 
sense  of  the  copula.  One  of  the  premises  in  each  case 
in  the  second  and  third  figures,  and  both  in  the  first, 
would  necessarily  be  in  the  inverse  order,  affirming  or 
denying  of  the  species  that  it  comprehends  or  does  not 
comprehend  the  genus,  or  else  the  original  signification 
of  the  copula  would  have  to  be  considered  as  reinstated 
in  such  premises,  and  the  process  would  thereby  lose 
its   distinctive   character    as   a   process  wholly  in   the 


SOEiTES. 


descending  direction,  which  only  we  are  now  considering. 
By  examining  the  synopsis,  it  will  be  fonnd  that  in  all 
cases  in  which  either  of  the  involved  Syllogisms  in  the 
columns  on  the  left  side  of  the  regressives  or  right  side 
of  the  progressives  is  in  one  of  the  imperfect  figures, 
and  in  all  cases  of  combinations  of  Syllogisms  shown 
on  the  other  side  of  each  page  respectively,  the  process 
of  the  reasoning  partakes  of  both  characters,  being  partly 
in  the  ascending  and  partly  in  the  descending  direction. 
I  shall  not  proceed  further  with  the  consideration 
of  this  subject,  for  the  reason  that  propositions  in  the 
descending  process  are  seldom,  if  ever,  put  forth  in  form 
as  herein  given,  but  in  the  converse.  When  you  come  to 
the  study  of  Logic  as  illustrated  by  concrete  examples 
(in  which  aspect  it  is,  in  respect  to  each  such  illustration, 
an  applied  science),  you  will  find  a  distinction  made  in 
respect  to  the  quantity  of  concepts  (terms)  as  being  either 
in  extension  or  intension,  the  latter  being  called  also 
comprehevsion.  This  distinction  runs  also  into  the  prop- 
ositions and  syllogisms  as  treated  of,  according  as  the 
terms  are  considered  as  in  one  or  the  other  quantity. 
You  will  find  it,  however,  to  be  of  no  practical  impor- 
tance in  so  far  as  the  process  of  reasoning  is  concerned,, 
all  reasoning  being  conducted  on  the  lines  of  the  pro- 
cess, as  we  have  previously  considered  it,  and  being 
called  reasoning  in  extension,  in  contradistinction  to  the 
process  as  shown  in  this  section,  which  is  called  reason- 
ing in  intension  or  comprehension.  The  distinction,  in 
so  far  as  it  relates  to  the  terms  (concepts),  does  not  lie 
within  the  province  of  Logic  as  a  Pure  Science,  and 
cannot   be   illustrated  by  means  of  symbols  indefinite 


74  LOGIC   AS   A   PUEE   SCIENCE. 

in  material  signification,  but  the  illustration  of  the  pro- 
cesses of  investigation  and  reasoning  wholly  in  the 
descending  direction,  given  in  this  section,  will  serve  to 
make  it,  as  continued  into  the  reasoning  process,  clearer 
and  more  easily  understood. 

The  consideration  of  the  subject  matter  of  this  sec- 
tion would  perhaps  have  been  more  appropriately  intro- 
duced when  treating  of  simple  Syllogisms,  but  it  could 
not  have  been  made  as  intelligible  without  as  with  geo- 
metrical illustration  by  combinations  of  triangles,  and 
the  latter  has  been  more  apj^ropriately,  and  at  the 
same  time  more  effectively,  introduced  in  this  chapter, 
where  it  has  been  exhibited  in  one  view,  and  to  its  full 
extent. 

The  copula  must  now  be  considered  as  returned  to 
its  original  signification,  and  where  the  word  "  descend- 
ing" shall  be  hereinafter  used,  it  must  be  considered 
as  applicable  to  the  direction  of  the  process  of  inves- 
tigation, but  not  to  the  form  of  the  propositions,  which, 
in  the  perfect  moods  of  the  Sorites,  will  always  be  found 
in  the  converse  of  those  herein  given. 

§  16.  Thus  far  the  premises  of  the  Sorites  exhibited 
have  consisted  of  propositions  put  forth  independently 
as  the  results  of  investigation.  They  may,  however,  be 
the  results  of  prior  processes  of  reasoning,  the  premises 
of  which  may  be  required  to  be  exhibited  in  connection 
with  them,  in  order  to  a  clear  understanding  of  the  prin- 
cipal argument.  The  full  expression  in  such  case  will 
become  complex,  and  may  be  in  two  forms,  of  which  I 
first  exhibit  the  following: 


SORITES. 

J  -  X, 

*.'  Z  —  X  and  Y  —  z  and  J  —  y. 

D-J. 

•.•  B  -  j  and  D  -  b. 

N  — a. 

•.•  K-d  and  N-k; 

N— X. 

• 

7a 


Here  each  premise  is  the  ultima  or  conclusion  of  a 
i^rior  process  of  reasoning,  the  premises  of  which  are 
affixed^  with  the  word  "because''  preceding. 

In  the  example,  all  the  premises  have  supporting 
X^remises  affixed.  But  any  one,  or  two,  only,  may  have 
such  premises  affixed,  the  other  two,  or  one,  as  the  case 
may  be,  being  propositions  put  forth  independently. 

The  whole  expression,  in  either  case,  is  called  an 
Ex^icheirema,  or  Reason-rendering  Syllogism  (of  either 
three  or  four  temis).  The  principal  argument,  with  ref- 
erence to  the  supi)orting  premises,  is  called  an  Episyllo- 
gism ;  and  the  supporting  premises  in  each  case,  ^vith 
reference  to  the  premise  proved,  is  called  a  Prosyllogism. 

The  second  form  is  that  in  which  the  premises  of  the 
Prosyllogism  are  prefixed^  those  in  relation  to  the  first 
premise  being  stated  antecedently  to  the  whole  principal 
expression :  those  in  relation  to  the  second  or  middle 
premise,  interpolated  between  the  first  and  middle,  and 
those  in  relation  to  the  last,  interpolated  between  the 
middle  and  last. 

If  either  of  the  first  two  be  in  such  form,  it  will  be 
found  ui)on  trial,  that  the  principal  expression  has  lost 
in  forcibleness  of  statement  or  in  perspicuity,  and  they 
may,  therefore,  be  disregarded,  but  the  third  \\ill  be 
found  to  lead  to  greater  perspicuity,  and  especially  so  if 
more  than  two  new  middle  terms  are  called  into  requi- 
sition for  the  purpose  of  elucidation. 


76  LOGIC   AS   A   PUKE   SCIENCE. 

The  first  form  (Epicheirema)  is  better  adapted  to  the 
statement  of  arguments  in  which  the  premises  are  ex- 
plained, the  second  to  those  in  which  either  the  first  or 
last  premise  is  disputed.  It  is  seldom  the  case  in  any 
disputation,  that  more  than  one  of  the  premises  of  the 
principal  argument  is  called  in  question,  and  that  one 
is  generally  the  first  or  last,  the  middle  premise  being 
usually  a  general  rule  acquiesced  in  upon  being  stated  ; 
and  if  the  disputed  yjremise  be  the  first,  the  principal 
argument,  by  changing  the  configuration,  may  be  thrown 
into  such  form  that  it  shall  become  the  last. 

I  now  proceed  to  consider  Sorites  as  complex  exj)res- 
sions,  in  the  second  form,  but  only  as  limited  to  those  in 
which  the  last  premise  is  disputed,  and  to  distinguish 
them  as  such,  shall  call  them  Compound  Sorites. 

§  17.  A  Compound  Sorites,  once  compounded,  tcTien 
fully  expressed,  consists  of  a  simple  Sorites  (herein 
called  the  principal  Sorites)  with  two,  or  three,  proposi- 
tions interpolated  between  its  middle  and  last  premises  ; 
such  propositions  (if  there  be  two)  constituting  the  pre- 
mises of  a  simple  Syllogism  of  which  the  conclusion,  or 
(if  there  be  three)  of  a  simple  Sorites,  of  which  the 
ultima  is  the  last  premise  of  \\\%  principal  Sorites.  The 
interpolated  propositions  will  be  herein  called  the  in- 
cluded Enthymeme,  if  there  be  two,  or  Sorites,  if 
there  be  three,  giving  the  full  name  in  the  latter 
case,  in  default  of  one  analogous  to  Enthymeme  in  the 
former.  An  included  Sorites  may  in  like  manner 
have  an  Enthymeme  or  second  Sorites  included  within 
it,  and  the  second  included  Sorites  may  in  like  manner 
have  an  Enthymeme  or  third  Sorites  included  within  it,. 


Ed 


SORITES.  77 

cind  so  on  ad  infinitum.  There  can  be  but  one  included 
Enthymeme,  and  it  will  always  be  the  last  included  ex- 
pression. The  reasoning  in  all  such  cases,  while  it  will 
ve  the  appearance  of  being  very  much  involved,  will 
in  reality  be  very  much  clearer. 

§  18.  But  compound  Sorites  are  seldom,  if  ever,  f^llly 
expressed  informal,  prepared  argumentation,  the  last 
premise  of  the  principal  Sorites  being  suppressed,  but, 
as  will  be  hereinafter  shown,  in  all  cases  implied.  In 
this  aspect  a  compound  Sorites  may  be  better  defined 
as  an  argument  consisting  of  more  than  four  expressed 
propositions  composed  of  as  many  terms  as  there  are 
expressed  propositions,  including  the  ultima.  Both 
definitions  Avill  be  better  understood  by  illustration. 

Let  us  suppose  the  case  of  two  disputants  of  whom 
one,  the  proponent,  advances  these  propositions : 

I>  -  J: 
.-.  X  -  X, 

to  which  the  other,  the  opponent,  answers  :  I  admit  that 
D  —  j,  but  I  deny  that  it  follows  that  X  —  x. 

The  propositions,  as  you  will  observe,  constitute  the 
abridged  form  of  the  first  mood. 

The  proponent  replies,  asserting,  as  the  reason,  the 
two  i)ropositions  necessary  to  make  up  the  expanded 
form,  viz.: 

•.•  J  —  X 
and  X  —  d, 

and  to  this  the  opponent  makes  rejoinder:   Admitting 
that  J  — -  X,  I  deny  that  X  —  d. 


78  LOGIC   AS   A   PURE   SCIENCE. 

The  issue  is  now  clearly  defined,  and  the  whole  case 
may  be  stated  as  follows  : 


J     —    X 

admitted, 

I)-  j 

admitted. 

X  —  d 

alleged  but  denied. 

X    —   X 

claimed  but  denied. 

The  proponent,  to  maintain  the  issue  on  his  part, 
must  establish  that  'N  —  d,  or  must  fail. 

To  do  it,  as  the  proposition  to  be  established  is  A,  he 
must  find  a  middle  term,  with  which  both  the  terms  'N 
and  D  may  be  compared,  so  as  to  form,  with  the  con- 
clusion, a  perfect  Syllogism  in  Barbara  (symbols  AAA), 
or  two  middle  terms,  with  one  of  which  IS"  may  be  com- 
pared and  D  with  the  other,  and  one  of  which  may  be 
predicated  of  the  other,  all  in  such  manner  as  to  con- 
stitute, with  the  ultima,  a  valid  Sorites  in  the  first  mood 
(symbols  A  AAA). 

Let  the  middle  term,  in  the  first  case,  be  Y,  and  the 
two  middle  terms,  in  the  second  case,  be  Y  and  Z. 

The  Syllogism  in  the  first  case  will  be  : 

Y-  d, 

N  -  y; 
.-.  N  —  d. 

But  in  the  second  case  the  two  new  terms  are  required 
to  be  compared,  and  either  may  be  the  subject  of  the 
proposition  in  which  they  are  compared,  viz.,  Y  —  z  or 
Z  —  y.     The  abridged  Sorites  may  therefore  be  either  : 

Y  —  z ;      .-.    N  —  d  ; 
or,  Z   -  y;      .-,   N  -  d. 


I 


SORITES.  79' 

Let  us  take  the  first,  and  in  order  to  expand  it  into 
a  full  Sorites,  let  ns  write  down  the  first  mood  in  the 
regressive  configuration,  as  in  the  synopsis,  and  write 
under  its  second  and  fourth  propositions  the  abridged 
Sorites  thus  taken,  as  follows  : 

J  —  x;    D  —  j;    N  —  d;      .-.  X  —  x. 
Y  —  z;  ,-.  :N^  —  d. 

Then^  by  expressing  the  first  and  third  implied 
propositions  of  the  abridged  Sorites  (making  them  to 
correspond  in  respect  to  the  terms  employed),  we  shall 
have  the  expanded  Sorites  as  follows : 

Z  —  d  ;    Y  —  z  ;    N  —  y;     .-.  N  —  d. 

By  taking  from  the  Syllogism  in  the  first  case  its  two 
premises  (constituting  an  Enthymeme  of  the  third  order), 
and  from  the  Sorites  in  the  second  case,  its  three  pre- 
mises, and  interpolating  them  (respectively)  between  the 
middle  and  last  premises  of  the  principal  Sorites,  w^e 
shall  have,  in  each  case,  a  compound  Sorites  fully  ex- 
pressed, as  follows : 


In  the  first  case. 

In  the  second  case. 

J    -    X, 

J    —    X, 

D-  j, 

D-  j> 

Y-d, 

Z  -d. 

N-y; 

Y-  2, 

.-. 

N  —  y; 

N  —  d. 

.•. 

and  .••  N  —  x. 

N-d, 

and  .-.  N  —  X. 

60  LOGIC  AS  A   PURE   SCIENCE. 

The  conclusion  of  the  first  Enthymeme  of  the  princi- 
pal Sorites,  viz.,  J)  —  x,  is  held  in  the  mind  ready  to 
unite  with  the  last  premise,  N  —  d  (after  the  latter  shall 
have  been  proved),  in  establishing  the  ultima,  N  —  x. 

§  19.  But  there  is  a  shorter  and  simpler  process,  and 
the  one  which  is  usually  employed  in  formal,  prepared 
argumentation.  Instead  of  holding  in  the  mind  the  con- 
clusion of  the  first  Enthymeme  to  unite  with  the  last 
premise  of  the  principal  Sorites  when  proved,  as  above 
stated,  we  may  at  once  employ  it  (mentally)  as  a  premise 
in  connection  with  the  first  of  the  new  expressed  proposi- 
tions, and  in  like  manner  the  unexpressed  conclusion  re- 
sulting from  them  as  a  premise  in  connection  with  the 
second  new  expressed  proposition  (and  in  the  second  case 
as  above,  the  unexpressed  conclusion  thus  resulting  in 
connection  with  the  third),  and  shall  find  that  the  last 
premise  of  the  principal  Sorites  will  not  appear.  Thus, 
in  the  two  cases,  the  unexpressed  conclusions  being  given 
in  italics : 

In  the  first  case.  In  the  second  case. 

J   —  X,  J    —  X, 

D   —    j,     (,.-.D-x).  J)   —    ],     (.:D-x). 

Y  —  d,   (.-.  Y-x).  Z  —  d,  (.-.  z-x). 

N  —  j;  Y  —  z,   (.-.  Y-x). 

.:  K  -  X.  N  -y; 

.-.  N  —  X. 

But  the  last  premise  of  the  principal  Sorites  will  have 
been  implied,  as  will  be  manifest  from  a  comparison  of 
the  two  forms  in  the  second  case  put  side  by  side,  as 
follows : 


soe!tes.  81 

First  form  i)i  second  case.  Second  form  in  second  case. 


J    -    X, 

J    —   X. 

D   —    j;    {..  B—x,  held  in  the  mind). 

D-  j, 

_  _  _  _ 

Z  —  d, 

Z  -  rt. 

Y  -  z, 

Y-  z, 

N  -  y ; 

N-y; 

.-.  N  —  X. 

"   X-d; 

aud  . 

•.    N  —  X.    (•••  D-x). 

The  second  form  is  the  simpler,  but  the  first  is  the 
clearer,  exhibiting  the  entire  process  of  the  reasoning. 

The  included  Enthymeme  in  the  first  case,  or  Sorites 
in  the  second,  serves  only  to  pi^ove  the  last  premise  of 
the  principal  Sorites,  and  forms  no  part  of  the  argument^ 
which  is  wholly  contained  in  the  principal  Sorites. 

§  20.  Both  the  principal  and  included  Sorites  in  the 
examples  are  in  the  regressive  configuration,  but  they 
may  be  in  different  configurations.  If  in  the  foregoing 
disputation  the  opponent  in  his  rejoinder  had  admitted 
the  magnus  premise,  !N^  —  d,  but  denied  the  maximus, 
J  —  X,  the  principal  Sorites  of  the  proponent  would 
have  been  in  the  progressive  configuration,  and  the  in- 
cluded one  could  have  still  been  in  the  regressive,  viz. : 


N  —  d, 

D  -  i; 

Z  -  X, 

Y  -z, 

J  -y; 

J     X, 

and 

.-.   N  —  X. 

82  LOGIC   AS  A   PURE   SCIENCE. 

The  two  configurations  cannot  be  directly  linked 
together  in  this  example,  as  before  shown  in  the  second 
form,  there  being  a  break  in  the  chain  between  the 
second  and  third  propositions.  But  by  considering  the 
configuration  of  the  included  Sorites  to  be  changed  (as 
it  may  be  by  transposing  the  first  and  third  premises 
thereof),  the  whole  expression  can  be  put  in  the  second 
form  as  before,  and  the  last  premise  of  the  principal 
Sorites,  J  —  x,  will  not  appear.  It  does  not,  however, 
follow  that  the  two  configurations  cannot  in  any  case  be 
directly  linked  together.  That  they  may  be  in  some 
cases  mil  be  hereinafter  seen. 

§  21.  All  the  Syllogisms  involved  in  all  the  foregoing 
examples  are  in  Barbara,  and  the  dictum  of  Aristotle, 
as.  hereinbefore  extended,  may  be  directly  applied  to 
those  in  the  second  form,  by  extending  it  still  further  in 
like  manner.  But  to  those  in  the  first  form  it  would 
have  to  be  twice  applied,  first  to  the  included  Sorites 
and  secondly  to  the  principal,  and  in  that  case  would 
not  require  to  be  further  extended,  both  the  Sorites 
being  simple. 

§  22.  But  if  any  of  the  involved  Syllogisms  are  in 
any  other  figure,  or  combination  of  figures,  they  would 
have  to  be  converted  into  Syllogisms  in  the  first  figure, 
before  the  dictum  could  be  directly  applied. 

The  following  are  examples  of  compound  Sorites, 
the  involved  Syllogisms  of  which  are  in  combinations  of 
figures,  as  shown  by  the  names  of  the  moods  given  in 
connection  with  them.  The  conclusions  proved,  but  not 
expressed,  are  also  given  in  italics  in  connection  with 
the  names  of  the  moods,  except  the  ultima  of  the  in- 


SORITES. 


83 


eluded  Sorites  (in  the  first  forms),  which  is  expressed  as 
a  premise  below  the  second  dotted  line.  The  principal 
Sorites  and  the  number  of  its  mood  and  the  configuration 
are  given  in  advance  of  each  example :_ 

Gth  Regressive  Mood. 
J  —  X,  d  —  j,  D  —  n ;    .-.    n  —  x. 


First  form. 
J  — X, 
d  —  j  ;     (.-.  d-x,  Dani), 

D— Z, 

Z-y, 
Y-n; 

D  — n; 

and    .*.    11  —  X.      (•.•  d—x,  DiBomis). 


Second  form. 
J-x, 

d  —  j,  (.-.  d—x.  DarU\ 

D  —  Z,  (.-.  z—x,  Di8amis)y 

7i  —  y,  (,'.  y—x,  Disamis\ 

Y-n; 

;.   n  —  X.      {Disamis). 


15th  Pi'ogressive  Jlood. 


N,  D— j,  X  — d;    .-.    ^  — X. 


First  form. 
D  — j;   (A  N-D, 
Y-d, 

z-y, 

X  — z: 


Second  form. 
^— N, 

D  —  j,  (.-.  ^—D,  Camenes), 

Y  —  d,  (.-.  N—  Y,  Came8treg\ 

Z  —  y,  (.♦.  ^—  Z,  Camestres\ 
X-z; 

*.  Jf — X.  iCamegtreg). 


X-d; 
and   /.    ?^  —  X.     (:•  N-D,  Camestres). 


84 


LOGIC   AS   A  PURE   SCIENCE. 


25th  Regressive  Mood. 


and 


d-^X, 

D 

-  i,  J  - 

—  n ;    .-.    n  -^  X. 

Nrst  form. 

Second  form. 

d— X, 

d-^X, 

D-j;    u.j- 

-X, 

Bokardo), 

D  —  j,     i.'.j-^X,  Bokardo\ 



J  — y,     {.-.y-^X,  Bokardo) 

J-y, 

Y  —  Z,     (.-.  z-^X,  Bokardo) 

Y-z, 

Z— n; 

Z-n; 

/.   n  -w  X.    {Bokardo). 

J-n; 

n-^X.    (vi- 

■  X,  Sokardo). 

%  23.  The  included  Sorites  may  have  an  Enthymeme 
or  a  second  Sorites  included  within  it,  and  the  second 
included  Sorites  may  have  an  Enthymeme  or  third  Sorites 
included  within  it,  and  so  on  ad  infinitum.     Thus : 


First  form 

Second  form. 

J-x, 

J  X, 

D  —  j  ;    {.-.  B  —  x,  held  in  t?is  mind). 

D  — j,    (.-.  D-x), 



Y-d,    (.-.  Y-x), 

j     Y-d, 

Z— y,    (.-.  z-x). 

(         Z  —  y;    (.-.  Z  — rf,  hdd  in  the  mind). 

z  —  k,    (.-.  k-x). 

1 



K  — q,    (.-.  q-x\ 

i 

z— k,  ^ 
K— q, 
Q  — n; 

^  Second  included  Sorites^ 

Q-n; 
.-.  n  —  X. 

(       z— n; 

( .-. (•••  Z-d), 

d— n; 

ar 

id  .-.  n  —  X. 

C-  D-x). 

SOEITES. 


85 


If  the  first  included  Sorites  in  the  last  example  be 
put  in  the  regressive  configuration,  its  last  premise  will 
be  Y  —  d  instead  of  z  —  n,  and  the  second  included 
Sorites  will  be  employed  to  establish  the  former  instead 
of  the  latter,  but  of  course  by  different  premises.  In 
such  case  we  shall  find  that  when  we  attempt  to  put  the 
whole  expression  in  the  second  form,  the  premises  of  the 
second  included  Sorites  will  take  precedence  of  those  in 
the  first,  and  the  latter  wiU  be  transposed.     Thus: 


M 


First  form. 
J  — X, 
13  —  J  •    (.'.  D  —  x,  held  in  the  mind), 

Z  —  n, 

Z  —  V  ;    (.-.  y  —  n,  held  in  the  mind)y 


K-q 
Y  — k 


Y-d; 

i-'y-ni 

d  — 11 ; 
and  .*.  n  — x.   (•.•  d-x). 


Second  form. 
J-x, 
D  — j,     i.'.D-x), 

Q  — d,  (.-.  Q-x), 

K  —  q,    (.-.  ^-a;), 

Y  — k,  (.'.  T-x), 
Z— y,  (.-.  z-x\ 
z  — n  : 


The  argumentation  is  supposed,  of  course,  to  have 
taken  place  on  the  lines  of  the  process  in  the  first  form, 
and  the  second  included  Sorites  did  not  therefore  come 
into  the  process  until  the  proposition,  Y  —  d,  was  dis- 
puted. The  illustration  thus  shows  the  superiority  of 
the  first  over  the  second  form,  as  exhibiting  the  whole 


OO  LOGIC   AS  A  PUKE  SCIENCE. 

process  of  the  reasoning.  The  second  could  not  have 
been  framed  until  the  first  had  been  gone  through  with. 

§  24.  Compound  Sorites  may,  however,  be  exhibited 
in  forms  which  at  first  sight  may  seem  to  be  in  contra- 
vention of  what  has  been  before  laid  down,  but  ujjon 
examination  it  will  be  found  that  such  is  not  the  case. 

Thus,  in  the  two  following  cases :  "^ 


(1.) 

(3.) 

■  N  — 

d,                                         N  -  d, 

D  — 

J,                                  D  -  h 

^  — 

X,                                   J  —  X, 

Y  — 

X,                                   ¥  -  X, 

Z  — 

y,                              z  -  y, 

Q  - 

z;                                      Q  —  z; 

/.  ^  — 

Q.                                '.i^-  Q. 

Let  us  take  the  second  and  write  in  line  with  each 

premise  (except 

the  first  and    last)  the  implied   con- 

elusions : 

N  —  d, 

D    —    j,     (.-.  ]!f-j), 

J    —    X,    i.'.N-X), 

Y  —  X,  (. .  ^-  D, 

Z  —  y,   (.-.  ^-z), 

Q  -  z; 

.'. 

5^  -  Q. 

The  expression,  with  the  exception  of  the  ultima,  will 
be  found,  upon  examination,  to  constitute  the  premises 
of  two  simple  Sorites,  of  which  the  first  is  in  the  pro- 
gressive configuration  and  the  second  in  the  regressive. 

♦  Taken  from  Schuyler's  Logic,  p.  88. 


SORITES. 


87 


By  stating    them    successively  with    their   implied 
ultimas,  we  shall  have  them  in  the  following  fonn : 


■ 

N—  d, 

1 

D-  J, 

1 

J    —   X  ;    (.-.  N—  X,  held  in  the  mind). 

1 

¥  —  X, 

i- 

z  —  y^ 

1 

Q  —    Z  ;     (.-.  -^-X,  held  in  the  mind). 

and  then. 

•/  N  —  X, 

and  ^  —  X; 

.-.  ^-N; 

or, 

•'  ^  -  X, 

and  M  —  x; 

/.   N  -  Q. 

I  now  proceed  to  show  that  Sorites,  stated  as  above, 
fall  within  the  definition  of  compound  Sorites,  as  herein- 
before given. 

The  maximus  premise,  being  the  last  premise  oi  the 
principal  Sorites  involved  in  the  foregoing  examples, 
has  not  appeared,  but  has  in  all  cases  been  implied. 
The  middle  premise  is  (as  has  been  before  stated  to  be 
the  case  in  all  Sorites,  simple  and  compound)  the  second, 
and  in  the  examples  is  D  —  j.  Combining  this  with  the 
ultima,  the  abridged  form  of  the  principal  Sorites  is 
therefore, 

D-  j; 

/.  ?f  —  Q. 


e8  LOGIC   AS  A   PURE  SCIENCE. 

Expanding  this  in  the  12th  progressive  mood  as  in 
the  synopsis,  we  shall  have  the  full  principal  Sorites  as 
follows : 

N_d,  D-j,  J-Q;    .'.    ^-Q; 

and  the  compound  Sorites  will  be  as  follows  : 


N  - 

d, 

D  — 

j; 

J  — 

X, 

¥  — 

X 

Z  — 

y. 

Q- 

z; 

(.•.  N—j,  held  in  the  mind). 


X  ;     (•••  ^—  y,  held  in  the  mind). 


^  -  Q; 

and  .*.  N  —  Q.      ( .•  n^-j). 


§  25.  But  the  magnus  and  maximus  terms  of  the 
principal  Sorites,  at  the  ultima  of  which  we  first  arrive, 
may  not  be  the  infima  species  and  summum  genus,  and 
further  investigation  may  bring  into  the  process  of  the 
reasoning  lower  species  or  higher  genera,  and  if  in  both 
directions,  both ;  and  the  new  term  or  terms,  instead  of 
being  employed  interiorly  as  middle  terms  as  hitherto, 
will  be  employed  exteriorly.  In  such  case  the  new 
term,  or  terms,  will  constitute,  if  there  be  but  one, 
a  new  magnus,  or  maximus  term  ;  or  if  there  be  two, 
obtained  by  investigation  in  both  directions,  both,  and 


SORITES.  89 

the  displaced  terms  will  become  middle  terms.  We  shall 
then  find  that  there  will  be  two  new  abridged  and  full 
principal  Sorites  in  each  case,  one  regressive  and  one 
progressive,  but  varying  according  as  the  new  term,  or 
terms,  are  applied  to  the  original  Sorites  considered  as 
both  regressive  and  progressive.  They  will,  however,  be 
independent  of  each  other,  and  each  will  have  its  correla- 
tive in  their  respectively  opposite  configumtions.  The 
displaced  original  term,  if  it  shall  have  been  the  magnus, 
will  become  the  minor-middle  term  of  the  new  principal 
progressive  Sorites,  and  will  not  appear  in  the  new  re- 
gressive, but  if  the  magnus  term  be  again  displaced  by 
bringing  in  another,  then  the  displaced  original  term  will 
become  the  major-middle ;  but  if  the  displaced  original 
term  shall  have  been  the  maximus,  then  it  will  become  the 
major-middle  term  of  the  new  principal  regressive  Sorites, 
and  will  not  appear  in  the  new  progressive,  and  if  the 
maximus  term  be  again  displaced  by  bringing  in  another, 
the  displaced  original  will  become  the  minor-middle  term. 

But  of  the  original  premises  in  the  case  of  one  new 
tenn  being  brought  in,  one,  or  two,  wall  still  remain  in 
each  new  principal  Sorites,  one  in  the  regressive  configu- 
ration, and  two  in  the  progressive,  if  the  new  term  be 
maximus,  and  vice  versa,  if  magnus.  One  original  premise 
only  will  remain  in  each  of  the  new  principal  Sorites  in 
any  case  if  two  new  terms,  one  magnus  and  one  max- 
imus, are  brought  in. 

The  original  ultima  will  of  course  have  disappeared  in 
every  case.  But  if  two  new  terms  are  brought  in,  both 
having  been  discovered  in  a  process  of  investigation  in 
one  direction  only,  the  original  ultima  will  reappear  as  a 


•90  LOGIC   AS   A   PUEE   SCIENCE. 

'premise  of  one  of  the  new  principal  Sorites,  the  regres- 
sive, if  the  investigation  were  in  the  ascending  direction, 
and  the  progressive,  if  in  the  descending. 

If  the  investigation  shall  be  pursued  so  that  more  than 
two  new  terms  shall  be  brought  in,  in  each  direction, 
every  vestige  of  the  original  principal  Sorites  will  have 
disappeared  from  the  new  principals,  as  they  will  then 
be  constituted. 

But  all  the  premises  of  the  original  principal  Sorites 
will,  in  all  cases,  be  found  to  remain,  either  partly  in  the 
principal  Sorites,  and  partly  in  the  following  included 
Enthymeme  or  Sorites,  or  in  two  of  the  included  Sorites, 
or  wholly  in  the  last  included  Sorites,  or  partly  in  the 
Enthymeme,  which  is  the  final  expression,  and  partly  in 
the  next  j)receding  included  Sorites,  according  as  the 
new  terms  shall  be  brought  in  ;  and  they  will  always  be 
found  together  in  their  original  order,  either  regressive 
or  progressive,  how  far  soever  the  process  be  continued, 
and  this,  also,  whether  the  compound  Sorites  be  in  the 
first  or  second  form,  as  hereinbefore  shown. 

The  following  examples  illustrate  all  the  foregoing 
remarks,  except  the  last,  as  to  compound  Sorites  in  the 
second  form,  which  can  be  verified  by  trial.  All  the  in- 
volved Syllogisms  are  in  the  first  figure  throughout. 

The  original  premises  and  ultima  (employed  as  a 
premise)  are  printed  in  Roman  letters,  and  those  which 
remain  in  the  j)rincipal  Sorites  in  full-faced  type.  All 
other  propositions  are  printed  in  Italics.  The  examples 
having  the  same  number  of  new  terms  are  so  arranged, 
either  on  the  same  or  opposite  pages,  that  they  may  be 
readily  compared. 


SORITES.  91 

With  one  neio  term,  brought  in  in  the  ascending  process  of  in- 
vestigation, and  therefore  a  new  maximus  term: 


Regressive  Configuration. 
J  _  X  :    (••  J- 

-yh 

Progressive  ConfigurcUion. 
N-d, 
D  —   j  ;  (.-.  N-j) 

K  —  d; 

* '                -J 

J     —    X, 

N-  j, 
and  .-.  X  —  y.    {.■  J- 

y)' 

J  —  y, 

and  .-.  N  —  y.  (•.•  n-J). 

Full  forms  of  neio  principal  Sorites : 

^—y,   J  — X,  iV^— y;    .-.    N—y. 
X_cl,  D  — j,   J  —  y.    :.   N  —  y. 

With  one  neio  term,  brought  in  in  the  descending  process,  and 
therefore  a  new  magnns  term: 


Regressive  Configuration. 

Progressive  Configuration. 

J  —  X, 

K  —  n, 

D  —  j:    i.-.D- 

-ar). 

X  —  d;(.-.  JT-flO. 

N-  d, 

D-  j, 

A'  —  n ; 

J   -  x; 

K—  d\ 

I)  —  x\ 

and  .-.  K —  x.  (•.•/>- 

x\ 

and 

:.  K  —  .r.  c-  K-d). 

Full  forms  of  new  principal  Sorites : 

J  — X,    D  — j,     K—d\    .'.   K—x. 
K—n,     N— d,    D—x\    .-.    K—x. 


9^  LOGIC   AS  A  PURE  SCIENCE. 


With  tivo  new  terms,  one  brought  in  in  the  ascending  process  of 
investigatiouy  and  therefore  a  neio  maximus  term,  and  the 
other  brought  in  in  the  descending  process,  and  therefore 
a  new  magnus  term: 


Regressive  Configurat 

ion. 

Progressive  Configuration. 

X-  y, 

K  —  /^, 

J   —  x;  (.-. 

J- 

-y\ 

N  —  d;  {.-.K-d), 

D  -  j, 

^  -  j. 

N  —  d. 

J     —    X, 

K  —  n\ 

^^-//; 

,', 

. 

.. 

A'-i; 

D  -  y. 

and  /.  K  —  y.   (•.• 

J- 

-y)- 

and  . 

\  K  —  y.   {,-K-d). 

Full  forms  of  neio  principal  Sorites : 


X—y,    J  —X,    £^—  j  ;    ■ 

:    K-y- 

K  —  n,     N  — d,    D—y;    . 

:  K-y. 

SORITES. 


93 


With  two  new  terms,  both   brought  in  in  the  ascending  process 
of  investigation,  and  07ie  therefore  a  new  maximus  term : 

Regressive  Configuration. 

Y  —  z, 

X  —   lj\{.-.  X-z), 

J    —  X, 

D  -  J, 

N  —  d; 

y  —  x; 

and  .-.  X  —  z.  (•.•  x—z). 

Full  forms  of  new  principal  Sorites : 


ressive 

Configuration. 

N 

-  d, 

D 

—  J;  (-• 

.  iV^- 

-J), 

J 

—   X, 

X 

—  y^ 

r 

—    Z'y 

J 

—    ^ 

/.  X 

—  Z.  C 

•  N- 

-J)- 

Y-  z, 

X-y, 

X— x: 

.'.    X—z. 

N— cl, 

D-j, 

J  —  z: 

.:   X  —  z. 

With  two  new  terms,  both  brought  in  in  the  descending  process^ 
and  one  therefore  a  neio  magnus  term: 


and 


\ive  Configuration, 

Progressive  Configuratioii. 

J-    X, 

Q-k, 

B  -  j;  (••■ 

D- 

-ar), 

K  —  n;  (.-.  $-n), 

N  —  d, 

N  —  d, 

K —  n, 

D-J, 

Q-k; 

J  —  x; 

Q-d; 

X-  x; 

Q  —  ■^.  (••• 

D- 

■x-). 

and  . 

•.    Q  —  X.    {:■  §-»). 

Full  forms  of  new  principal  Sorites  : 

J— X,    D  — j,      Q  —  d;    .'.    Q  —  x. 

Q  —  k,    K  —  n,     N  —  X ;    .-.    Q  —  x. 


94 


LOGIC   AS  A  PURE  SCIENCE. 


With  three  new  terms,  of  which  two  are  brought  in  in  the  ascend- 
ing process  of  investigation,  and  one  of  them  therefore  a  neio 
maximus  term,  and  the  third  in  the  descetiding  process, 
and  therefore  a  7iew  magnus  term: 


Regressive  Configuration. 

r-  z, 


Progressive  Configuration. 
K  —  n, 
N  —  d;   i.'.K-d\ 


1      D 


j  ;   i.-.D-x), 


{.'. {:■  J)-x), 

K  —  X', 
and  .*.  K  —  z.    c-  x-2). 


Full  forms  of  new  principal  Sorites : 
Y—z,    X—y,    K—x',    .'.   K—z, 


SORITES. 


95. 


With  three  new  terms,  of  which  two  are  brought  in  iji  the  desce7icl- 
i7ig  process  of  investigation,  and  one  of  them  therefore  a  new 
magnus  term,  and  the  third  in  the  ascending  process,  and  i 
therefore  a  new  maximus  term: 


Regressive  Configuratimh. 

J   —  x;    (.-.  J-y\ 


J   I)  -  J. 

i      N  —  (1;   {.-.N-j), 


K 

Q 
Q 

and  .-.  Q 


Progressive  Configuration. 


N-S). 


J    X, 


{... 


J  —  y\ 

(viV-i), 

N  —  y; 
and  .-.  Q  —  y.     C-  G-»X 


Full  forms  of  new  principal  Sorites : 

X—y,  J  — X,    Q  —  j', 
Q-^lc,  K—n,   N—y, 


Q-y- 
Q-y- 


96 


LOGIC   AS  A  PURE  SCIENCE. 


With  four  netv  terms,  of  which  two  are  brought  in  in  the  ascend- 
ing process,  and  tivo  in  the  descending  : 


Regressive  Configuration. 
Y  —  z, 
X  —  y\  (.-.  X- «), 


Progressive  Configuration. 

Q  -h 

K  —  n\   (.-.  q-n\ 


/       J    -  X, 


N  —  d, 
K  —  n, 


f      Q  -  d', 

Q  -x; 
and  .-.  Q  —  z.     c  x-  z). 


J      N-d, 

I      D  —  j;    (.-.  iVr-A 


J  — 

x/ 

X  — 

y. 

Y  — 

^; 

_  _  - 

-  -  , 

{... 


J  —  z; 

N  —  z; 
and  .-.  Q  —  z.     (.-  Q  —  n). 


Full  forms  of  new  principal  Sorites : 

Y-  z,    X-y,    Q-x;    .-.    Q  -  z. 

Q  —  h,    K  —  n,    N  —  z\    :.    Q  —  z. 


SORITES. 


97 


With  eight  neiu  terms,  of  which  four  are  brought  in  in  the 
ascending  process,  and  four  in  the  descending: 

Progressive  Configuration. 


Regressive  Configuration. 
S  —    t, 
Z  —  s\    (.-.  z  -  0, 

I      Y  -  z, 

i  .  X  —  y\    (.-.  x-z\ 


s  J 


3;     (.-.V-x), 


j     X  -  d, 

l     K  —  n :   (.-.  K-  d). 


Q- 

h' 

G  - 

q^ 

H  - 

■9'^ 

— 

—  . 

y .-. (V  D-x), 

i   .*. (•••  X-  2), 

H  —  z\ 

and  .-.  H  —  i.  ( .•  z  -  0. 


(.      K  —  n-.  (.-.  Q  - «), 


J  N  -  d, 

I  D  —  j;    (.-.  ivr-i), 

/  J    -  X, 

i  X  —  y;    i.-.J-y), 


r-  z,^ 

Z  —  s, 

S  —  t; 


j     r  -  t; 

I  .'. (•••  J-y), 


1 

J    — 

t; 

t. 

(•• 

•  A'- 

-S). 

{. 

X  — 

t; 

*. 

(•• 

•  Q- 

-n), 

Q  - 

t; 

and  . 

.H  — 

t. 

(•. 

H- 

-?)■ 

Full  forms  of  new  principal  Sorites : 

S-t,     Z—s,     H—z',    .'.   H  —  t. 
H-g,     G--q,     Q-t',    .-.   H-t. 


98 


LOGIC   AS   A   PUEE   SCIENCE. 


§  36.  To  recur  now  to  illustration  by  means  of  geo- 
metrical figures. 

A  regular  tetrahedron  may  by  four  sections,  beginning 
in  the  middle  of  each  of  its  edges  and  made  parallel  to 
the  opposite  planes  respectively,  be  divided  into  five  fig- 
ures, of  vrhich  four  v^ill  be  regular  tetrahedra,  and  the 
fifth  and  interior  figure  a  regular  octahedron. 

Thus,  by  reproducing  our  former  illustration  on  card- 
board before  folding,  and  dividing  it  by  lines  which 
shall  represent  the  four  sections,  we  shall  have  the 
following  : 


ISTow,  assuming  each  interior  dotted  line  to  be  the 
edge  of  an  equilateral  triangular  plane,  represented  by 
card-board,  projecting  backward,  divergingly,  at  the 
proper  dihedral  angles,  from  the  plane  of  the  one  which 
we  are  supposed  to  have  in  hand,  then,  by  folding  the 
latter  as  before,  we  shall  have  a  combination  of  five 
figures,  as  above  stated,  which  will  present  to  our  eyes 
successively,  as  we  turn  it  about  as  before,  the  following 
figures : 


SORITES. 

First  Series. 


N\  /X 

Second  Series. 


Each  of  the  four  tetrahedra  having  one  original  exte- 
rior point,  and  three  visible  and  one  invisible  planes,  will 
be  found  to  have  that  point  marked  with  one  of  the  let- 
ters N,  D,  J,  X  on  each  visible  plane ;  the  fifth  figure, 
the  octahedron,  having  no  original  exterior  point,  and 
four  visible,  and  four  invisible  planes,  will  be  found 
marked  on  each  visible  plane  vdth  one  of  the  numbers 
1,  2,  3,  4.  It  is  wholly  included,  and  occupies  all  the 
space,  between  the  invisible  planes  of  the  four  tetra- 
hedra and  planes  connecting  their  visible  planes,  and 
its  volume  is  exactly  equal  to  the  sum  of  their  vol- 
umes ;  and  it  may  well  be  regarded  as  the  analogue 
of  the  ultima  conclusio  of  the  Sorites,  of  which  the 
abridged  form  is : 

D-j;  .-.  N 


X. 


The  analogy  between  a  compound  Sorites  in  which  the 
original  principal  Sorites  shall  remain  the  principal,  and 


100  LOGIC   AS   A  PUEE  SCIENCE. 

a  Sorites  be  interpolated  as  hereinbefore  shown,  and 
a  tetrahedron  divided  by  sections  as  represented  in  the 
foregoing  illustration,  cannot  be  exhibited  as  simply  or 
as  clearly  as  that  between  a  simple  Sorites  and  a  tetra- 
hedron considered  as  a  unit,  as  in  our  former  illustration, 
because  the  tetrahedron  which  is  the  analogue  of  the 
included  Sorites  is  involved  in  and  forms  an  indistin- 
guishable, but,  as  must  be  regarded,  separate,  part  of 
the  included  octahedron,  having  one  of  the  visible  planes 
of  the  octahedron  as  its  only  visible  face.  Its  invisible 
faces  cannot  be  brought  to  the  surface  in  the  following 
figures,,  but  must  be  regarded  as  represented  by  the  three 
triangles  by  which  its  visible  face  is  bounded,  the  ultimate 
point  of  which  will  be  found  marked  X  in  the  figures. 
Its  ultimate  point  will  not  be  the  point  X  as  shown  in 
the  figures,  but  will  lie  in  the  perpendicular  let  fall  from 
the  point  N  upon  the  opposite  plane  of  the  original 
te-trahedron.  We  shall  hereinafter  find  that  perpendic- 
ular to  be  part  of  one  axis  of  a  sphere  produced  by  the 
revolution  of  the  tetrahedron,  and  that  the  pole  of  that 
axis  opposite  N  should  be  marked  X.  The  ultimate 
point  of  the  indistinguishable  tetrahedron  which  is  the 
analogue  of  the  included  Sorites,  may  be  at  any  point 
in  the  line  of  this  axis  within  the  octahedron,  and  let  us 
assume  that  point  to  be  in  the  centre  of  the  invisible 
plane  of  the  octahedron  opposite  its  visible  plane  which 
is  the  visible  face  of  the  involved  tetrahedron.  The  in- 
visible faces  of  the  latter  will  then  be  equal  to  the  tri- 
angles by  which  its  visible  face  is  bounded  in  the  figures. 
Let  us  suppose  that  in  the  progressive  process  we 
have  established  the  relation  between  N  and  J,  as  in  the 
lower  one  of    the  following   combination   of    triangles 


SORITES.  101 

(which,  observe,  are  the  same  as  the  triangles  1  and  3 
in  our  original  card-board  illustration),  and  that  the 
relation  between  J  and  X  requires  to  be  established. 


We  shall  then  have  the  upper  triangle  in  which  only  the 
relation  (length  of  line)  between  D  and  J  is  known,  and 
let  us  suppose  that  the  relation  between  each  of  those 
l^oints  and  X  is  not  capable  of  being  immediately  deter- 
mined, but  that  there  are  two  points  {middle  terms\  one 
in  each  of  the  other  two  lines,  capable  of  being  succes- 
sively reached  from  D  or  any  point  in  the  line  B  J  ex- 
cept the  point  J,  and  the  length  of  a  straight  line  con- 
necting them  capable  of  being  measured,  and  from  both 
of  which  the  direction  of  X  can  be  observed,  and  the 
angles  therefore  determined. 

Keproducing  the  upper  triangle  and  marking  the  mid- 
dle point  in  the  base  line  J',  and  the  two  points  at  the 
extremities  of  the  base  X'  and  X",  and  the  two  new  points 
Y  and  Z  at  the  middle  of  each  of  the  two  lines  connect- 
ing the  extremes  of  the  base  with  X,  and  connecting  such 
new  points,  and  each  of  them  with  J',  we  shall  have  the 
following : 


102 


LOGIC   AS   A  PUEE   SCIENCE. 


and  we  may  now  establish  that  J'  —  X  in  the  same  man- 
ner as  we  have  hereinbefore  established  that  N  —  X. 

But,  the  lines  X'  X  and  X"  X  are,  by  construction, 
equal  to  J  X  and  D  X  in  the  upper  triangle,  on  the 
preceding  page,  the  middle  points  in  which  may  be 
marked  Z  and  Y.  In  the  process  we  have  found  J'  Z 
equal  to  J'  Y,  and  X'  Z  equal  to  J'  Z.  X'  Z  is  therefore 
equal  to  J'  Y.  But  X'  Z  is  J  Z.  And  as  J'  Y  is  equal 
to  J'  Z,  it  will,  upon  being  applied  to  the  latter,  coincide 
with  it,  and  the  point  Y  will  fall  upon  the  point  Z. 
J  Z  may  therefore  be  called  J  Y,  and  is  equal  to  J'  Y. 

The  whole  combination  of  triangles  will  now  be  as  fol- 
lows, the  original  letters  being  put  on  the  outside  : 


SORITES.  103 

We  can  now  express  the  full  compound  Sorites,  ex- 
emplified by  the  foregoing  illustration,  as  follows  : 

X  -  d, 

D    —    j:   {.'.  X—j,  held  in  the  mind), 

J    —  y.     (=  J  Z  =  J'  Y,  fi-om  which  latter  directum  of  X odserred), 
Y  —    Z,     <=T  Z,  relation,  i.  e.,  length  of  line  known), 
Z    —   X  :   {=  Z  X,  direction  observed  from  former), 

J   -  X  ; 
and  .'.  X  —  X.    (•.•x—j). 

This  is  the  same  compound  Sorites  as  that  exhibited 
in  §  20,  on  page  81,  but  with  the  included  Sorites  in  the 
progressive,  instead  of  the  regressive,  configuration. 

But  if  the  interpolated  expression  be  an  Enthymeme, 
the  analogy  will  be  much  clearer,  as  the  lines  by  which 
the  Enthymeme  will  be  represented  wiU  lie  wholly  in  the 
surface  and  not  involve  any  section  of  the  original  figure. 

Thus,  if  in  the  following  combination  of  triangles 
(which  observe  are  the  same  as  triangles  3  and  4  in  our 
original  card -board  illustration) : 


we  shall,  in  like  manner  as  before,  have  established  the 
relation  between  N"  and  J  (as  in  the  upper,  left-hand 
triangle^  from  which  latter  we  can  see  X,  but  are 
unable  immediately  to  determine     its  distance,    with- 


104 


LOGIC   AS   A   PURE  SCIENCE. 


out  the  knowledge  of  which  we  cannot  establish  the 
relation  between  N  and  X ;  we  may  select  another  me- 
diate point,  Y,  which  can  be  reached,  and  distance 
measured  from  J,  and  from  which  X  may  also  be  seen, 
and  the  angles  therefore  determined,  as  in  the  following 
figure : 


and  then,  by  the  elements  thus  obtained,  we  can  deter- 
mine the  required  distance  from  J  to  X,  and  by  means 
thereof  and  the  elements  previously  obtained,  the  dis- 
tance from  N"  to  X. 

The  compound  Sorites  exemplified  by  the  foregoing 
illustration  will  be  as  follows : 


N  —  d, 

D  -  j ;  (.• 

iV  — j  held  in  the  mind), 

Y-  X, 

J  -y; 

J     -X, 

.-.  N  —  X.    c 

N-j). 

But  if,  instead  of  having  begun  in  the  ascending 
direction,  we  shall  have  begun  in  the  descending,  and 
have  established  the  relation  between  X  and  D,  as  in 
the  lower,  right-hand  one  of  the  following  combination  of 
triangles  (1  and  2  in  the  figure  on  page  65) : 


SORITES. 


106. 


and  shall  then,  although  able  to  see  N  from  D,  but  not 
from  X,  be  unable  to  determine  its  distance  from  D, 
without  the  knowledge  of  which,  it  would  be  impossible 
to  determine  its  distance  from  X  ;  we  may,  in  like  man- 
ner as  before,  select  another  mediate  point  K,  which  can 
be  reached  from  D,  and  from  which  N  can  also  be  seen, 
as  in  the  following  figure  : 


and  then,  as  before,  may  determine  the  required  distance 
from  D  to  N,  and  by  means  thereof  and  the  elements 
previously  obtained,  the  distance  from  X  to  N. 

The  compound  Sorites  exemplified  by  the  foregoing 
illustration,  will  be  as  follows  : 


X   comprehends   J, 

J    comprehends   D  J    (. 

J)   comprehends   K, 
K  comprehends   N  J 


X  comprehends  A  ^d  in  the  mind), 


D   comprehends   N  j 
and   .'.    X   comprehends    N.     {'.' X  amiprehends  B). 


106  LOGIC   AS  A   PURE  SCIENCE. 

By  putting  together  the  first  of  each  of  the  two  sets 
of  figures  in  the  preceding  illustrations,  on  the  line  D  J, 
common  to  both,  we  shall  have  the  following  figure : 


which  is  the  same  as  that  on  page  61,  but  in  a  different 
position.  By  turning  triangle  2  downward  in  a  semi- 
circle on  the  point  D  as  a  centre,  we  shall  have  our 
original  card-board  figure  ;  or  by  turning  triangle  4  up- 
ward to  the  like  extent  on  the  point  J  as  a  centre,  we 
shall  have  the  figure  shown  on  page  65.  Triangles  1  and  2 
taken  together  and  3  and  4  taken  together  are  analogues 
of  progressive  Sorites,  1  and  2,  in  the  descending  direction, 
and  3  and  4,  in  the  ascending ;  but  if  2  and  4  be  both 
turned  as  above  described,  they  will  become  analogues  of 
regressive  Sorites  in  the  respectively  opposite  directions. 

§  27.  All  the  four  triangles  in  our  original  card -board 
illustration  are  equilateral  and  equal.  The  solid  figure 
resulting  from  the  folding  of  the  card-board  is  a  regular 
tetrahedron,  which  is  defined  as  a  solid  having  four 
faces,  all  equal  equilateral  triangles.    But  the  triangles 


SORITES.  107 

raight  have  been  all  equal  isosceles  triangles,  or  partly 
equilateral  and  partly  isosceles.  Such  can  be  exhibited 
in  a  plane  figure  bounded,  by  three,  or  four  exterior  lines, 
if  the  triangles  are  all  equal,  or  by  six,  if  they  are  partly 
equilateral  and  partly  isosceles,  and  capable  of  being 
folded  so  that  the  exterior  points  shall  meet  in  a  perfect^ 
but  not  regular^  figure.  But  a  'perfect  tetrahedron  may 
have  all  its  faces  unequal,  and  in  such  case  the  faces  may 
be  spread  out  in  an  irregular  plane  figure  having  ^\q 
exterior  lines.  In  all  cases  the  number  of  exterior  lines 
will  be  found  to  be  six,  if  bisected  lines  are  counted  each 
as  two.  All  other  plane  figures  having  all  the  points 
exterior  are  imperfect  and  cannot  be  folded,  so  that  the 
exterior  points  will  meet,  and  their  areas,  and  conse- 
quently the  volume  of  space  which  they  can  be  made 
resjiectively  to  inclose,  can  only  be  determined  by  means 
of  the  triangle.  Imperfect  Syllogisms  and  Sorites  in 
logic  must  be  reduced  to  the  perfect  figure  before  they 
can  be  submitted  to  the  dictum  de  omni  et  nullo. 

§  28.  On  the  other  hand,  a  tetrahedron  (regular  or 
perfect)  may  be  added  to  on  the  outside  by  superimpos- 
ing on  each  of  its  faces  another  tetrahedron  having  a 
similar  face,  so  that  there  shall  be  ^ve  tetrahedra  in 
all.  Four  new  points  vrill  have  been  added,  all  exterior 
to  the  original  figure,  the  original  x^oints  becoming  in- 
terior, but  their  locations  visible,  the  original  figure 
having  otherwise  wholly  disappeared  from  view. 

Similarly,  as  we  have  before  seen,  in  respect  to  a 
Sorites,  when  four  new  terms  have  been  brought  in  ex- 
teriorly, two  in  each  direction,  the  four  propositions  of 


108  LOGIC    AS   A   PUKE   SCIENCE. 

the  original  principal  Sorites  will  have  disappeared  from 

the  two  new  principals^  as  they  will  then  be  constituted, 

but  they  will  remain  in  the  included  Sorites,  of  which 

the  inner  tetrahedron  is  the  analogue. 

But  in  the  figure,  formed  as  above  described,  the  four 

new  points,  which  we  will  consider  as  marked  K,  Q,  Y, 

and  Z,  will  furnish  only  one  new  principal  Sorites,  as  its 

analogue,  which  may  be  rendered  in  its  abridged  form 

thus  y 

V   Q-y;    /.  K-z. 

But  observe,  the  interior  figure  in  the  foregoing  com- 
bination is  a  tetrahedron,  not  necessarily  regular,  but  per- 
fect ;  and  if,  instead  of  beginning  with  such  a  one,  con- 
sidered as  a  unit,  we  begin  with  a  regular  one  considered 
as  divided  by  four  sections,  as  before  shown,  and  super- 
pose upon  each  of  the  visible  planes  of  the  included 
octahedron,  a  tetrahedron  similar  to  each  of  the  four 
resulting  from  such  sections,  we  shall  have  a  solid  figure 
in  the  form  of  an  eight-pointed  star,  the  octahedron 
having  entirely  disappeared  from  view,  except  that  the 
locations  of  its  points  will  be  visible.  This  eight-pointed 
star  will  be  found  to  consist  of  two  equal  intervolved  regu- 
lar tetrahedra,  to  both  of  which  the  interior  octahedron 
will  be  common,  and  its  revolution  about  its  centre  will 
produce  a  sphere  exactly  equal  to  that  produced  by  the 
revolution  of  the  original  tetrahedron.  Four  exterior 
points  will  have  been  added,  but  of  these  two  are  the 
opposite  poles  of  the  two  original  points  marked  N  and 
X,  and,  having  a  common  relation  with  them  to  the  in- 
cluded octahedron,  should  be  marked  X  and  N  respect- 
ively, leaving,  in  fact,  but  two  new  independent  points, 


SORITES.  109 

which  may  be  marked  Y  and  Z.     The  whole  figure  will 

then  be  the  analogue  of  two  independent  full  Sorites,  of 

which  one  only  is  new,  and  that  only  in  part,  the  abridged 

forms  being : 

•.•  D  —  j ;  .-.  N  —  X. 

*.•  Y  —  z  ;   .-.  N  —  X. 

Bj'  comparing  the  foregoing  illustrations  with  the 
Sorites  having  four  new  terms  added  exteriorly,  given 
on  page  96,  the  superiority  of  the  Sorites  over  its  ana- 
logue, the  tetrahedron,  will  again  be  manifest. 

§  29.  Thus  everywhere,  whether  we  go  inwardly  or 
outwardly,  and  in  all  things,  metaphysical  as  well  as 
physical,  we  find  triniunity,  and  can  thence  proceed  to 
quadriunity,  but  beyond  that,  except  in  composite 
forms,  we  cannot  go. 

§  30.  From  the  foregoing  definitions  and  illustrations 
of  Sorites,  simple  and  compound,  it  seems  manifest  that 
the  human  mind  is  limited  to  reasoning  concerning  the 
relations  of  four  terms.  If  other  terms  are  brought  in, 
they  must  relate  to  the  terms  of  the  principal  argument, 
and  in  such  case,  if  such  relation  be  to  the  middle  terms, 
they  serve  only  to  elucidate,  but  if  to  the  magnus  and 
maximus  terms,  then  they  supplant  those  terms ;  which, 
if  there  be  one,  or  two  successively  of  each  (new  magnus 
and  maximus  terms)  respectively,  become  terms  of  the 
two  new  middle  premises  respectively,  but  if  more 
than  two  of  each,  then  are  relegated  to  the  subordi- 
nate position  of  middle  terms  employed  only  in  elucida- 
tion. Otherwise  they  must  be  the  terms  of  independent 
arguments. 


110  LOGIC   AS   A   PURE   SCIENCE. 

§  31.  There  remains  but  to  say  that  I  have  not  pointed 
out  the  characteristics  of  Sorites,  nor  given  the  rules  in 
relation  to  them,  as  the  same  have  been  usually  pointed 
out  and  given  (or  in  part  so)  in  logical  treatises,  and  to 
which  reference  has  been  hereinbefore  made  ;  and  I  now 
refer  to  them  only  for  the  purpose  of  showing  their 
inadequacy. 

They  have  been  written  with  reference  to  Sorites 
treated  of  as  capable  of  being  expanded  only  in  Syllo- 
gisms wholly  in  the  first  figure^  and  without  reference,  of 
course,  to  the  distinction  between  them  as  simple  and 
compound,  which  has  been  hitherto  unobserved.  They 
relate, 

1st.  To  the  number  of  Syllogisms  involved,  as  equal 
to  the  number  of  middle  terms,  and  as  ascertainable  from 
the  number  of  premises  of  the  Sorites,  less  one. 

2d.  To  the  character  of  the  premises  of  the  involved 
Syllogisms,  whether  minor  or  major,  and  the  number  of 
each  and  their  sequence,  viz.:  one  only,  and  that  the 
first,  major,  and  all  the  following  minor  in  a  regressive 
Sorites  ;  and  mce  versa^  in  a  progressive. 

3d.  To  the  number  and  positions  of  particular  and 
negative  premises  in  the  tw^o  configurations,  viz.  :  that 
one  only  can  be  particular,  and  that  the  last,  and  one 
only  negative,  and  that  the  first,  in  a  regressive  Sorites ; 
and  vice  versa  (in  respect  to  positions)  in  a  progressive. 

The  first  is  true  of  all  Sorites,  simple  and  compound, 
in  respect  to  the  number  of  Syllogisms  involved  being 
equal  to  the  number  of  middle  terms,  and  has  been  im- 
pliedly shown  as  true  of  all  simple  Sorites,  in  respect  to 
such  number  being  ascertainable  from  the  number  of 


SOillTES.  Ill 

premises  less  one,  in  that  they  have  been  described  as 
having  three  premises,  and  as  being  capable  of  expan- 
sion into  two  Syllogisms ;  but  in  such  latter  respect  it 
does  not  apply  to  compound  Sorites  when  fully  ex- 
pressed. 

The  second,  by  an  examination  of  the  synopsis,  will 
be  found  to  hold  good,  of  all  regressive  simple  Sorites  in 
respect  to  the  moods  in  which  they  are  minors,  and  not 
good  in  respect  to  those  in  which  they  are  majors,  and 
mce  versa  of  all  progressives. 

The  third  is  of  course,  and  for  obvious  reasons,  appli- 
cable to  all  simple  Sorites  (but  not  to  all  compound^ 
when  fully  expressed)^  so  far  as  the  number  of  particu- 
lar and  negative  premises  is  concerned,  but  to  state  it  in 
respect  to  their  positions  as  applicable  to  all  Sorites 
capable  of  being  expanded  in  Syllogisms  ickolly  in  the 
first  figure^  and  also  to  some  in  combinations  of  figures, 
either  partly  or  not  at  all  of  that  figure^  and  then  to 
point  out  the  very  numerous  exceptions  in  other  like 
cases,  would  tend  rather  to  confuse  than  to  enlighten ; 
and  I  therefore  leave  the  subject,  and  pass  on  to  the  con- 
sideration of  Fallacies. 


U^   0?  THE        -<*- 


^. 


*l 


I 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

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